# 6 1

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 6 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 6 1 at Knotilus! 6_1 is also known as "Stevedore's Knot" (see e.g. [1]), and as the pretzel knot P(5,-1,-1).

 A Kolam of a 3x3 dot array 3D depiction Polygonal depiction Simple square depiction An other one Necklace

### Knot presentations

 Planar diagram presentation X1425 X7,10,8,11 X3948 X9,3,10,2 X5,12,6,1 X11,6,12,7 Gauss code -1, 4, -3, 1, -5, 6, -2, 3, -4, 2, -6, 5 Dowker-Thistlethwaite code 4 8 12 10 2 6 Conway Notation [42]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 7, width is 4,

Braid index is 4

[{8, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {7, 2}, {6, 8}, {1, 7}]
 knot 6_1. A graph, knot 6_1

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index ${\displaystyle \{3,4\}}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-5][-3] Hyperbolic Volume 3.16396 A-Polynomial See Data:6 1/A-polynomial

[edit Notes for 6 1's three dimensional invariants]
6_1 is a ribbon knot (drawings by Yoko Mizuma):

 a ribbon diagram isotopy to a ribbon
6_1 has two slice disks, by Scott Carter
Scott Carter notes that 6_1 bounds two distinct slice disks. He says: "this was spoken of in Fox's Example 10, 11, and 12 in a Quick Trip through Knot Theory ... BTW, the cover of Carter and Saito's Knotted Surfaces and Their Diagrams contains an illustration of such a slice disk". A picture is on the right.

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 0}$ Topological 4 genus ${\displaystyle 0}$ Concordance genus ${\displaystyle 0}$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial ${\displaystyle -2t+5-2t^{-1}}$ Conway polynomial ${\displaystyle 1-2z^{2}}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 9, 0 } Jones polynomial ${\displaystyle q^{2}-q+2-2q^{-1}+q^{-2}-q^{-3}+q^{-4}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle a^{4}-z^{2}a^{2}-a^{2}-z^{2}+a^{-2}}$ Kauffman polynomial (db, data sources) ${\displaystyle a^{3}z^{5}+az^{5}+a^{4}z^{4}+2a^{2}z^{4}+z^{4}-3a^{3}z^{3}-2az^{3}+z^{3}a^{-1}-3a^{4}z^{2}-4a^{2}z^{2}+z^{2}a^{-2}+2a^{3}z+2az+a^{4}+a^{2}-a^{-2}}$ The A2 invariant ${\displaystyle q^{14}+q^{12}-q^{6}-q^{4}+q^{-2}+q^{-6}+q^{-8}}$ The G2 invariant ${\displaystyle q^{66}+q^{62}-q^{60}+q^{56}-q^{54}+2q^{52}+q^{46}+q^{42}-q^{38}+q^{32}-2q^{28}+q^{26}+q^{24}-2q^{20}-2q^{18}+q^{16}-q^{14}+q^{12}-2q^{10}-q^{8}+2q^{6}-q^{4}-1+q^{-4}+q^{-10}+2q^{-14}-q^{-18}+q^{-20}+q^{-24}+q^{-28}+q^{-34}+q^{-38}}$

### "Similar" Knots (within the Atlas)

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

### Vassiliev invariants

 V2 and V3: (-2, 1)
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 ${\displaystyle -8}$ ${\displaystyle 8}$ ${\displaystyle 32}$ ${\displaystyle {\frac {116}{3}}}$ ${\displaystyle {\frac {52}{3}}}$ ${\displaystyle -64}$ ${\displaystyle -{\frac {304}{3}}}$ ${\displaystyle -{\frac {64}{3}}}$ ${\displaystyle -24}$ ${\displaystyle -{\frac {256}{3}}}$ ${\displaystyle 32}$ ${\displaystyle -{\frac {928}{3}}}$ ${\displaystyle -{\frac {416}{3}}}$ ${\displaystyle -{\frac {2791}{15}}}$ ${\displaystyle {\frac {884}{15}}}$ ${\displaystyle -{\frac {10084}{45}}}$ ${\displaystyle {\frac {343}{9}}}$ ${\displaystyle -{\frac {871}{15}}}$

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 ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$). The squares with yellow highlighting are those on the "critical diagonals", where ${\displaystyle j-2r=s+1}$ or ${\displaystyle j-2r=s-1}$, where ${\displaystyle s=}$0 is the signature of 6 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-4-3-2-1012χ
5      11
3       0
1    21 1
-1   11  0
-3   1   -1
-5 11    0
-7       0
-91      1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$