# K11a277

Jump to navigationJump to search

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a277 at Knotilus!

### Knot presentations

 Planar diagram presentation X6271 X10,4,11,3 X16,5,17,6 X12,8,13,7 X4,10,5,9 X18,11,19,12 X20,14,21,13 X22,16,1,15 X2,17,3,18 X8,19,9,20 X14,22,15,21 Gauss code 1, -9, 2, -5, 3, -1, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -6, 10, -7, 11, -8 Dowker-Thistlethwaite code 6 10 16 12 4 18 20 22 2 8 14

### Three dimensional invariants

 Symmetry type Reversible Unknotting number ${\displaystyle \{1,2\}}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a277/ThurstonBennequinNumber Hyperbolic Volume 16.9207 A-Polynomial See Data:K11a277/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial ${\displaystyle -t^{4}+6t^{3}-17t^{2}+28t-31+28t^{-1}-17t^{-2}+6t^{-3}-t^{-4}}$ Conway polynomial ${\displaystyle -z^{8}-2z^{6}-z^{4}-2z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{3,t+1\}}$ Determinant and Signature { 135, 2 } Jones polynomial ${\displaystyle -q^{8}+4q^{7}-8q^{6}+14q^{5}-19q^{4}+21q^{3}-22q^{2}+19q-13+9q^{-1}-4q^{-2}+q^{-3}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-10z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-10z^{2}a^{-2}+7z^{2}a^{-4}-2z^{2}a^{-6}+3z^{2}-4a^{-2}+2a^{-4}+3}$ Kauffman polynomial (db, data sources) ${\displaystyle 3z^{10}a^{-2}+3z^{10}a^{-4}+8z^{9}a^{-1}+15z^{9}a^{-3}+7z^{9}a^{-5}+7z^{8}a^{-2}+7z^{8}a^{-4}+8z^{8}a^{-6}+8z^{8}+4az^{7}-21z^{7}a^{-1}-40z^{7}a^{-3}-8z^{7}a^{-5}+7z^{7}a^{-7}+a^{2}z^{6}-39z^{6}a^{-2}-28z^{6}a^{-4}-9z^{6}a^{-6}+4z^{6}a^{-8}-23z^{6}-9az^{5}+18z^{5}a^{-1}+38z^{5}a^{-3}-10z^{5}a^{-7}+z^{5}a^{-9}-2a^{2}z^{4}+49z^{4}a^{-2}+31z^{4}a^{-4}-2z^{4}a^{-6}-6z^{4}a^{-8}+20z^{4}+2az^{3}-6z^{3}a^{-1}-6z^{3}a^{-3}+6z^{3}a^{-5}+3z^{3}a^{-7}-z^{3}a^{-9}-21z^{2}a^{-2}-11z^{2}a^{-4}+3z^{2}a^{-6}+2z^{2}a^{-8}-9z^{2}-za^{-1}-5za^{-3}-5za^{-5}-za^{-7}+4a^{-2}+2a^{-4}+3}$ The A2 invariant ${\displaystyle q^{8}-2q^{6}+3q^{4}+1+4q^{-2}-5q^{-4}+3q^{-6}-4q^{-8}+q^{-12}-3q^{-14}+4q^{-16}-q^{-18}+q^{-20}+q^{-22}-q^{-24}}$ The G2 invariant Data:K11a277/QuantumInvariant/G2/1,0

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

Same Alexander/Conway Polynomial: {K11a99,}

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

### Vassiliev invariants

 V2 and V3: (-2, -2)
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 ${\displaystyle -8}$ ${\displaystyle -16}$ ${\displaystyle 32}$ ${\displaystyle {\frac {116}{3}}}$ ${\displaystyle {\frac {76}{3}}}$ ${\displaystyle 128}$ ${\displaystyle {\frac {608}{3}}}$ ${\displaystyle {\frac {32}{3}}}$ ${\displaystyle 80}$ ${\displaystyle -{\frac {256}{3}}}$ ${\displaystyle 128}$ ${\displaystyle -{\frac {928}{3}}}$ ${\displaystyle -{\frac {608}{3}}}$ ${\displaystyle -{\frac {151}{15}}}$ ${\displaystyle -{\frac {172}{5}}}$ ${\displaystyle -{\frac {10204}{45}}}$ ${\displaystyle {\frac {1399}{9}}}$ ${\displaystyle -{\frac {1111}{15}}}$

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=}$2 is the signature of K11a277. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-4-3-2-101234567χ
17           1-1
15          3 3
13         51 -4
11        93  6
9       105   -5
7      119    2
5     1110     -1
3    811      -3
1   612       6
-1  37        -4
-3 16         5
-5 3          -3
-71           1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}}$ ${\displaystyle {\mathbb {Z} }^{11}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}}$ ${\displaystyle {\mathbb {Z} }^{11}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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

### Modifying This Page

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

Back to the top.