# 6 2

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 6 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 6 2 at Knotilus! Dror likes to call 6_2 "The Miller Institute Knot", as it is the logo of the Miller Institute for Basic Research.
 The Miller Institute Mug [1] Simple square depiction 3D depiction

### Knot presentations

 Planar diagram presentation X1425 X5,10,6,11 X3948 X9,3,10,2 X7,12,8,1 X11,6,12,7 Gauss code -1, 4, -3, 1, -2, 6, -5, 3, -4, 2, -6, 5 Dowker-Thistlethwaite code 4 8 10 12 2 6 Conway Notation [312]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 6, width is 3,

Braid index is 3

[{8, 2}, {1, 6}, {7, 3}, {2, 4}, {6, 8}, {3, 5}, {4, 7}, {5, 1}]
 Knot 6_2. A graph, knot 6_2.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index ${\displaystyle \{3,4\}}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-1] Hyperbolic Volume 4.40083 A-Polynomial See Data:6 2/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 1}$ Topological 4 genus ${\displaystyle 1}$ Concordance genus ${\displaystyle 2}$ Rasmussen s-Invariant -2

### Polynomial invariants

 Alexander polynomial ${\displaystyle -t^{2}+3t-3+3t^{-1}-t^{-2}}$ Conway polynomial ${\displaystyle -z^{4}-z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 11, -2 } Jones polynomial ${\displaystyle q-1+2q^{-1}-2q^{-2}+2q^{-3}-2q^{-4}+q^{-5}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{2}a^{4}+a^{4}-z^{4}a^{2}-3z^{2}a^{2}-2a^{2}+z^{2}+2}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{2}a^{6}+2z^{3}a^{5}-za^{5}+2z^{4}a^{4}-2z^{2}a^{4}+a^{4}+z^{5}a^{3}-za^{3}+3z^{4}a^{2}-6z^{2}a^{2}+2a^{2}+z^{5}a-2z^{3}a+z^{4}-3z^{2}+2}$ The A2 invariant ${\displaystyle q^{16}-q^{8}-q^{4}+q^{2}+1+q^{-2}+q^{-4}}$ The G2 invariant ${\displaystyle q^{86}-q^{84}+q^{82}-q^{80}-q^{78}-q^{74}+3q^{72}-2q^{70}+q^{68}+2q^{62}-q^{60}+q^{58}+q^{56}+q^{52}+3q^{46}-3q^{44}+q^{42}-q^{40}-q^{38}+2q^{36}-4q^{34}+q^{32}-2q^{30}-3q^{24}+q^{22}-q^{20}-q^{14}+q^{12}+q^{10}+2q^{6}-q^{4}+2q^{2}+1-q^{-2}+3q^{-4}-q^{-6}+2q^{-8}-q^{-12}+2q^{-14}+q^{-18}}$

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

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (-1, 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 ${\displaystyle -4}$ ${\displaystyle 8}$ ${\displaystyle 8}$ ${\displaystyle {\frac {34}{3}}}$ ${\displaystyle {\frac {38}{3}}}$ ${\displaystyle -32}$ ${\displaystyle -{\frac {208}{3}}}$ ${\displaystyle -{\frac {64}{3}}}$ ${\displaystyle -24}$ ${\displaystyle -{\frac {32}{3}}}$ ${\displaystyle 32}$ ${\displaystyle -{\frac {136}{3}}}$ ${\displaystyle -{\frac {152}{3}}}$ ${\displaystyle {\frac {2129}{30}}}$ ${\displaystyle {\frac {662}{15}}}$ ${\displaystyle -{\frac {1862}{45}}}$ ${\displaystyle {\frac {463}{18}}}$ ${\displaystyle -{\frac {751}{30}}}$

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 ${\displaystyle t^{r}q^{j}}$ 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 ${\displaystyle s=}$-2 is the signature of 6 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-4-3-2-1012χ
3      11
1       0
-1    21 1
-3   11  0
-5  11   0
-7 11    0
-9 1     -1
-111      1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$