# K11a367

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a367 at Knotilus! K11a367 is the next knot in the sequence trefoil, cinquefoil, septafoil, nonafoil... (See also T(11,2).) K13a4878 comes after it.

 Interlaced form of 11/2 star polygon or "undecagram" Decorative interlaced form of 11/2 star polygon or "undecagram" Decorative knotwork cross

### Knot presentations

 Planar diagram presentation X12,2,13,1 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X2,14,3,13 X4,16,5,15 X6,18,7,17 X8,20,9,19 X10,22,11,21 Gauss code 1, -7, 2, -8, 3, -9, 4, -10, 5, -11, 6, -1, 7, -2, 8, -3, 9, -4, 10, -5, 11, -6 Dowker-Thistlethwaite code 12 14 16 18 20 22 2 4 6 8 10

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 5 3-genus 5 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a367/ThurstonBennequinNumber Hyperbolic Volume Not hyperbolic A-Polynomial See Data:K11a367/A-polynomial

### Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus ${\displaystyle 5}$ Rasmussen s-Invariant -10

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{5}-t^{4}+t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3}-t^{-4}+t^{-5}}$ Conway polynomial ${\displaystyle z^{10}+9z^{8}+28z^{6}+35z^{4}+15z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 11, 10 } Jones polynomial ${\displaystyle -q^{16}+q^{15}-q^{14}+q^{13}-q^{12}+q^{11}-q^{10}+q^{9}-q^{8}+q^{7}+q^{5}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{10}a^{-10}+10z^{8}a^{-10}-z^{8}a^{-12}+36z^{6}a^{-10}-8z^{6}a^{-12}+56z^{4}a^{-10}-21z^{4}a^{-12}+35z^{2}a^{-10}-20z^{2}a^{-12}+6a^{-10}-5a^{-12}}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{10}a^{-10}+z^{10}a^{-12}+z^{9}a^{-11}+z^{9}a^{-13}-10z^{8}a^{-10}-9z^{8}a^{-12}+z^{8}a^{-14}-8z^{7}a^{-11}-7z^{7}a^{-13}+z^{7}a^{-15}+36z^{6}a^{-10}+29z^{6}a^{-12}-6z^{6}a^{-14}+z^{6}a^{-16}+21z^{5}a^{-11}+15z^{5}a^{-13}-5z^{5}a^{-15}+z^{5}a^{-17}-56z^{4}a^{-10}-41z^{4}a^{-12}+10z^{4}a^{-14}-4z^{4}a^{-16}+z^{4}a^{-18}-20z^{3}a^{-11}-10z^{3}a^{-13}+6z^{3}a^{-15}-3z^{3}a^{-17}+z^{3}a^{-19}+35z^{2}a^{-10}+25z^{2}a^{-12}-4z^{2}a^{-14}+3z^{2}a^{-16}-2z^{2}a^{-18}+z^{2}a^{-20}+5za^{-11}+za^{-13}-za^{-15}+za^{-17}-za^{-19}+za^{-21}-6a^{-10}-5a^{-12}}$ The A2 invariant ${\displaystyle q^{-18}+q^{-20}+2q^{-22}+q^{-24}+q^{-26}-q^{-42}-q^{-44}-q^{-46}}$ The G2 invariant ${\displaystyle q^{-90}+q^{-92}+q^{-94}+q^{-98}+2q^{-100}+2q^{-102}+q^{-104}+q^{-106}+2q^{-108}+3q^{-110}+2q^{-112}+q^{-116}+2q^{-118}+q^{-120}-q^{-124}+q^{-128}-2q^{-132}-q^{-134}-q^{-138}-q^{-140}-q^{-142}-q^{-144}-q^{-150}-q^{-188}-q^{-190}-q^{-196}-q^{-198}-q^{-200}-q^{-206}-q^{-208}+q^{-264}}$

### "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: (15, 55)
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 ${\displaystyle 60}$ ${\displaystyle 440}$ ${\displaystyle 1800}$ ${\displaystyle 4230}$ ${\displaystyle 610}$ ${\displaystyle 26400}$ ${\displaystyle {\frac {135344}{3}}}$ ${\displaystyle {\frac {23936}{3}}}$ ${\displaystyle 5368}$ ${\displaystyle 36000}$ ${\displaystyle 96800}$ ${\displaystyle 253800}$ ${\displaystyle 36600}$ ${\displaystyle {\frac {985183}{2}}}$ ${\displaystyle {\frac {70174}{3}}}$ ${\displaystyle 176338}$ ${\displaystyle {\frac {4919}{2}}}$ ${\displaystyle {\frac {44287}{2}}}$

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=}$10 is the signature of K11a367. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
01234567891011χ
33           1-1
31            0
29         11 0
27            0
25       11   0
23            0
21     11     0
19            0
17   11       0
15            0
13  1         1
111           1
91           1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=9}$ ${\displaystyle i=11}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=1}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=8}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=9}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=10}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=11}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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