# K11n4

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n4 at Knotilus!

 Knot K11n4. A graph, knot K11n4. A part of a knot and a part of a graph.

### Knot presentations

 Planar diagram presentation X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X18,12,19,11 X6,14,7,13 X15,20,16,21 X12,18,13,17 X19,22,20,1 X21,16,22,17 Gauss code 1, -5, 2, -1, 3, -7, 4, -2, 5, -3, 6, -9, 7, -4, -8, 11, 9, -6, -10, 8, -11, 10 Dowker-Thistlethwaite code 4 8 10 14 2 18 6 -20 12 -22 -16
A Braid Representative

### Three dimensional invariants

 Symmetry type Chiral Unknotting number ${\displaystyle \{1,2\}}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n4/ThurstonBennequinNumber Hyperbolic Volume 12.5531 A-Polynomial See Data:K11n4/A-polynomial

### Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus ${\displaystyle [0,3]}$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial ${\displaystyle -t^{3}+5t^{2}-11t+15-11t^{-1}+5t^{-2}-t^{-3}}$ Conway polynomial ${\displaystyle -z^{6}-z^{4}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 49, 0 } Jones polynomial ${\displaystyle q^{6}-3q^{5}+5q^{4}-7q^{3}+8q^{2}-8q+8-5q^{-1}+3q^{-2}-q^{-3}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{6}a^{-2}-4z^{4}a^{-2}+z^{4}a^{-4}+2z^{4}-a^{2}z^{2}-6z^{2}a^{-2}+2z^{2}a^{-4}+5z^{2}-a^{2}-3a^{-2}+a^{-4}+4}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{9}a^{-1}+z^{9}a^{-3}+5z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}+az^{7}+z^{7}a^{-1}+3z^{7}a^{-3}+3z^{7}a^{-5}-15z^{6}a^{-2}-8z^{6}a^{-4}+z^{6}a^{-6}-6z^{6}-az^{5}-8z^{5}a^{-1}-17z^{5}a^{-3}-10z^{5}a^{-5}+3a^{2}z^{4}+16z^{4}a^{-2}+3z^{4}a^{-4}-3z^{4}a^{-6}+13z^{4}+a^{3}z^{3}+5az^{3}+12z^{3}a^{-1}+16z^{3}a^{-3}+8z^{3}a^{-5}-3a^{2}z^{2}-10z^{2}a^{-2}-z^{2}a^{-4}+2z^{2}a^{-6}-10z^{2}-a^{3}z-3az-5za^{-1}-5za^{-3}-2za^{-5}+a^{2}+3a^{-2}+a^{-4}+4}$ The A2 invariant Data:K11n4/QuantumInvariant/A2/1,0 The G2 invariant Data:K11n4/QuantumInvariant/G2/1,0

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

Same Alexander/Conway Polynomial: {9_27, K11n21, K11n172,}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {9_41, K11n21,}

### 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 ${\displaystyle 0}$ ${\displaystyle -8}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 8}$ ${\displaystyle 0}$ ${\displaystyle {\frac {112}{3}}}$ ${\displaystyle {\frac {64}{3}}}$ ${\displaystyle 24}$ ${\displaystyle 0}$ ${\displaystyle 32}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 112}$ ${\displaystyle {\frac {200}{3}}}$ ${\displaystyle -{\frac {8}{3}}}$ ${\displaystyle {\frac {64}{3}}}$ ${\displaystyle -16}$

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 K11n4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-3-2-10123456χ
13         11
11        2 -2
9       31 2
7      42  -2
5     43   1
3    44    0
1   44     0
-1  25      3
-3 13       -2
-5 2        2
-71         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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