K11a288

K11a287

K11a289

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a288 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a288/ThurstonBennequinNumber Hyperbolic Volume 20.2067 A-Polynomial See Data:K11a288/A-polynomial

[edit Notes for K11a288's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 0

[edit Notes for K11a288's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-7t^{3}+23t^{2}-44t+55-44t^{-1}+23t^{-2}-7t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+z^{6}+z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 205, 0 }
Jones polynomial	$q^{6}-5q^{5}+12q^{4}-21q^{3}+29q^{2}-33q+34-29q^{-1}+22q^{-2}-13q^{-3}+5q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+4z^{6}-2a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}+7z^{4}-2a^{2}z^{2}-4z^{2}a^{-2}+z^{2}a^{-4}+6z^{2}-a^{2}-a^{-2}+3$
Kauffman polynomial (db, data sources)	$5z^{10}a^{-2}+5z^{10}+16az^{9}+28z^{9}a^{-1}+12z^{9}a^{-3}+20a^{2}z^{8}+21z^{8}a^{-2}+11z^{8}a^{-4}+30z^{8}+13a^{3}z^{7}-12az^{7}-45z^{7}a^{-1}-15z^{7}a^{-3}+5z^{7}a^{-5}+5a^{4}z^{6}-31a^{2}z^{6}-65z^{6}a^{-2}-21z^{6}a^{-4}+z^{6}a^{-6}-79z^{6}+a^{5}z^{5}-14a^{3}z^{5}-13az^{5}+6z^{5}a^{-1}-4z^{5}a^{-3}-8z^{5}a^{-5}-2a^{4}z^{4}+19a^{2}z^{4}+47z^{4}a^{-2}+14z^{4}a^{-4}-z^{4}a^{-6}+53z^{4}+6a^{3}z^{3}+10az^{3}+8z^{3}a^{-1}+8z^{3}a^{-3}+4z^{3}a^{-5}-6a^{2}z^{2}-12z^{2}a^{-2}-3z^{2}a^{-4}-15z^{2}-a^{3}z-2az-2za^{-1}-za^{-3}+a^{2}+a^{-2}+3$
The A2 invariant	$-q^{14}+3q^{12}-5q^{10}+3q^{8}+q^{6}-5q^{4}+8q^{2}-5+6q^{-2}-q^{-4}-2q^{-6}+5q^{-8}-6q^{-10}+3q^{-12}-2q^{-16}+q^{-18}$
The G2 invariant	Data:K11a288/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a288 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a288"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{4}-7t^{3}+23t^{2}-44t+55-44t^{-1}+23t^{-2}-7t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+z^{6}+z^{4}+z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 205, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{6}-5q^{5}+12q^{4}-21q^{3}+29q^{2}-33q+34-29q^{-1}+22q^{-2}-13q^{-3}+5q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+4z^{6}-2a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}+7z^{4}-2a^{2}z^{2}-4z^{2}a^{-2}+z^{2}a^{-4}+6z^{2}-a^{2}-a^{-2}+3$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$5z^{10}a^{-2}+5z^{10}+16az^{9}+28z^{9}a^{-1}+12z^{9}a^{-3}+20a^{2}z^{8}+21z^{8}a^{-2}+11z^{8}a^{-4}+30z^{8}+13a^{3}z^{7}-12az^{7}-45z^{7}a^{-1}-15z^{7}a^{-3}+5z^{7}a^{-5}+5a^{4}z^{6}-31a^{2}z^{6}-65z^{6}a^{-2}-21z^{6}a^{-4}+z^{6}a^{-6}-79z^{6}+a^{5}z^{5}-14a^{3}z^{5}-13az^{5}+6z^{5}a^{-1}-4z^{5}a^{-3}-8z^{5}a^{-5}-2a^{4}z^{4}+19a^{2}z^{4}+47z^{4}a^{-2}+14z^{4}a^{-4}-z^{4}a^{-6}+53z^{4}+6a^{3}z^{3}+10az^{3}+8z^{3}a^{-1}+8z^{3}a^{-3}+4z^{3}a^{-5}-6a^{2}z^{2}-12z^{2}a^{-2}-3z^{2}a^{-4}-15z^{2}-a^{3}z-2az-2za^{-1}-za^{-3}+a^{2}+a^{-2}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a288"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $t^{4}-7t^{3}+23t^{2}-44t+55-44t^{-1}+23t^{-2}-7t^{-3}+t^{-4}$, $q^{6}-5q^{5}+12q^{4}-21q^{3}+29q^{2}-33q+34-29q^{-1}+22q^{-2}-13q^{-3}+5q^{-4}-q^{-5}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(1, 0)

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 $4$ $0$ $8$ ${\frac {14}{3}}$ $-{\frac {14}{3}}$ $0$ $0$ $32$ $-32$ ${\frac {32}{3}}$ $0$ ${\frac {56}{3}}$ $-{\frac {56}{3}}$ ${\frac {511}{30}}$ ${\frac {898}{15}}$ $-{\frac {1858}{45}}$ $-{\frac {511}{18}}$ $-{\frac {449}{30}}$

V_2,1 through V_6,9 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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of K11a288. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     4   -4 9                   8 1   7 7                 13 4     -9 5               16 8       8 3             17 13         -4 1           17 16           1 -1         13 18             5 -3       9 16               -7 -5     4 13                 9 -7   1 9                   -8 -9   4                     4 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=0$ ${\mathbb {Z} }^{18}\oplus {\mathbb {Z} }_{2}^{16}$ ${\mathbb {Z} }^{17}$ $r=1$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{17}$ ${\mathbb {Z} }^{17}$ $r=2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{16}$ ${\mathbb {Z} }^{16}$ $r=3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a287 K11a289