K11a139

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K11a138.gif

K11a138

K11a140.gif

K11a140

Contents

K11a139.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a139 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X16,6,17,5 X18,8,19,7 X2,10,3,9 X20,11,21,12 X22,13,1,14 X8,16,9,15 X6,18,7,17 X14,19,15,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -8, 5, -2, 6, -11, 7, -10, 8, -3, 9, -4, 10, -6, 11, -7
Dowker-Thistlethwaite code 4 10 16 18 2 20 22 8 6 14 12
A Braid Representative
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A Morse Link Presentation K11a139 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a139/ThurstonBennequinNumber
Hyperbolic Volume 13.7305
A-Polynomial See Data:K11a139/A-polynomial

[edit Notes for K11a139's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant -2

[edit Notes for K11a139's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-12 t^2+20 t-23+20 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6-2 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 99, 2 }
Jones polynomial -q^8+3 q^7-6 q^6+10 q^5-14 q^4+16 q^3-15 q^2+14 q-10+6 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -6 z^6 a^{-2} +2 z^6 a^{-4} +z^6-14 z^4 a^{-2} +9 z^4 a^{-4} -z^4 a^{-6} +4 z^4-14 z^2 a^{-2} +13 z^2 a^{-4} -3 z^2 a^{-6} +5 z^2-4 a^{-2} +5 a^{-4} -2 a^{-6} +2
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +6 z^9 a^{-3} +3 z^9 a^{-5} +6 z^8 a^{-2} +7 z^8 a^{-4} +5 z^8 a^{-6} +4 z^8+3 a z^7-3 z^7 a^{-1} -10 z^7 a^{-3} +z^7 a^{-5} +5 z^7 a^{-7} +a^2 z^6-20 z^6 a^{-2} -21 z^6 a^{-4} -9 z^6 a^{-6} +3 z^6 a^{-8} -10 z^6-9 a z^5-5 z^5 a^{-1} +4 z^5 a^{-3} -12 z^5 a^{-5} -11 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+25 z^4 a^{-2} +29 z^4 a^{-4} +8 z^4 a^{-6} -6 z^4 a^{-8} +7 z^4+7 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} +17 z^3 a^{-5} +9 z^3 a^{-7} -2 z^3 a^{-9} +2 a^2 z^2-19 z^2 a^{-2} -19 z^2 a^{-4} -4 z^2 a^{-6} +2 z^2 a^{-8} -4 z^2-a z-2 z a^{-1} -3 z a^{-3} -5 z a^{-5} -3 z a^{-7} +4 a^{-2} +5 a^{-4} +2 a^{-6} +2
The A2 invariant q^8-q^6+2 q^4-q^2-1+2 q^{-2} -3 q^{-4} +4 q^{-6} - q^{-8} + q^{-10} + q^{-12} -2 q^{-14} +3 q^{-16} - q^{-18} - q^{-24}
The G2 invariant q^{46}-2 q^{44}+5 q^{42}-9 q^{40}+10 q^{38}-10 q^{36}+2 q^{34}+15 q^{32}-34 q^{30}+55 q^{28}-63 q^{26}+50 q^{24}-15 q^{22}-44 q^{20}+112 q^{18}-162 q^{16}+172 q^{14}-120 q^{12}+15 q^{10}+115 q^8-220 q^6+271 q^4-231 q^2+111+47 q^{-2} -192 q^{-4} +253 q^{-6} -214 q^{-8} +98 q^{-10} +53 q^{-12} -158 q^{-14} +179 q^{-16} -114 q^{-18} -25 q^{-20} +168 q^{-22} -258 q^{-24} +232 q^{-26} -109 q^{-28} -77 q^{-30} +265 q^{-32} -369 q^{-34} +357 q^{-36} -226 q^{-38} +23 q^{-40} +189 q^{-42} -339 q^{-44} +362 q^{-46} -257 q^{-48} +90 q^{-50} +97 q^{-52} -207 q^{-54} +215 q^{-56} -124 q^{-58} -9 q^{-60} +126 q^{-62} -179 q^{-64} +133 q^{-66} -18 q^{-68} -116 q^{-70} +219 q^{-72} -235 q^{-74} +174 q^{-76} -60 q^{-78} -77 q^{-80} +175 q^{-82} -225 q^{-84} +203 q^{-86} -128 q^{-88} +34 q^{-90} +55 q^{-92} -112 q^{-94} +129 q^{-96} -112 q^{-98} +72 q^{-100} -24 q^{-102} -18 q^{-104} +39 q^{-106} -46 q^{-108} +39 q^{-110} -24 q^{-112} +12 q^{-114} + q^{-116} -7 q^{-118} +7 q^{-120} -7 q^{-122} +4 q^{-124} -2 q^{-126} + q^{-128}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a57, K11a108, K11a231,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {}

Vassiliev invariants

V2 and V3: (1, 3)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
4 24 8 \frac{254}{3} \frac{58}{3} 96 272 -32 120 \frac{32}{3} 288 \frac{1016}{3} \frac{232}{3} \frac{34831}{30} -\frac{5062}{15} \frac{33062}{45} \frac{2225}{18} \frac{2671}{30}

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=2 is the signature of K11a139. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          2 2
13         41 -3
11        62  4
9       84   -4
7      86    2
5     78     1
3    78      -1
1   48       4
-1  26        -4
-3 14         3
-5 2          -2
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

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K11a138.gif

K11a138

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K11a140