# K11n83

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n83 at Knotilus!

### Knot presentations

 Planar diagram presentation X4251 X8394 X5,17,6,16 X7,12,8,13 X2,9,3,10 X11,18,12,19 X13,1,14,22 X15,21,16,20 X17,10,18,11 X19,7,20,6 X21,15,22,14 Gauss code 1, -5, 2, -1, -3, 10, -4, -2, 5, 9, -6, 4, -7, 11, -8, 3, -9, 6, -10, 8, -11, 7 Dowker-Thistlethwaite code 4 8 -16 -12 2 -18 -22 -20 -10 -6 -14
A Braid Representative

### Three dimensional invariants

 Symmetry type Reversible Unknotting number ${\displaystyle \{1,2\}}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n83/ThurstonBennequinNumber Hyperbolic Volume 12.0122 A-Polynomial See Data:K11n83/A-polynomial

### Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus ${\displaystyle [0,2]}$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial ${\displaystyle 3t^{2}-12t+19-12t^{-1}+3t^{-2}}$ Conway polynomial ${\displaystyle 3z^{4}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \left\{2,t^{2}+t+1\right\}}$ Determinant and Signature { 49, 0 } Jones polynomial ${\displaystyle q^{4}-3q^{3}+5q^{2}-7q+9-8q^{-1}+7q^{-2}-5q^{-3}+3q^{-4}-q^{-5}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{2}a^{4}+z^{4}a^{2}-a^{2}+2z^{4}+4z^{2}+4-3z^{2}a^{-2}-3a^{-2}+a^{-4}}$ Kauffman polynomial (db, data sources) ${\displaystyle az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+z^{8}a^{-2}+4z^{8}+4a^{3}z^{7}+az^{7}-3z^{7}a^{-1}+3a^{4}z^{6}-7a^{2}z^{6}-3z^{6}a^{-2}-13z^{6}+a^{5}z^{5}-10a^{3}z^{5}-5az^{5}+9z^{5}a^{-1}+3z^{5}a^{-3}-7a^{4}z^{4}+7a^{2}z^{4}+10z^{4}a^{-2}+z^{4}a^{-4}+23z^{4}-2a^{5}z^{3}+6a^{3}z^{3}+6az^{3}-7z^{3}a^{-1}-5z^{3}a^{-3}+2a^{4}z^{2}-4a^{2}z^{2}-11z^{2}a^{-2}-2z^{2}a^{-4}-15z^{2}-a^{3}z-az+za^{-1}+za^{-3}+a^{2}+3a^{-2}+a^{-4}+4}$ The A2 invariant Data:K11n83/QuantumInvariant/A2/1,0 The G2 invariant Data:K11n83/QuantumInvariant/G2/1,0

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

Same Alexander/Conway Polynomial: {9_41,}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {9_27,}

### Vassiliev invariants

 V2 and V3: (0, -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 ${\displaystyle 0}$ ${\displaystyle -8}$ ${\displaystyle 0}$ ${\displaystyle -32}$ ${\displaystyle -24}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {80}{3}}}$ ${\displaystyle {\frac {160}{3}}}$ ${\displaystyle -72}$ ${\displaystyle 0}$ ${\displaystyle 32}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 32}$ ${\displaystyle {\frac {232}{3}}}$ ${\displaystyle -24}$ ${\displaystyle -48}$ ${\displaystyle -32}$

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 K11n83. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-101234χ
9         11
7        2 -2
5       31 2
3      42  -2
1     53   2
-1    45    1
-3   34     -1
-5  24      2
-7 13       -2
-9 2        2
-111         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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