K11n81

K11n80

K11n82

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n81 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,16,6,17 X2837 X9,21,10,20 X11,1,12,22 X13,19,14,18 X15,6,16,7 X17,13,18,12 X19,15,20,14 X21,11,22,10
Gauss code	1, -4, 2, -1, -3, 8, 4, -2, -5, 11, -6, 9, -7, 10, -8, 3, -9, 7, -10, 5, -11, 6
Dowker-Thistlethwaite code	4 8 -16 2 -20 -22 -18 -6 -12 -14 -10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 4 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n81/ThurstonBennequinNumber Hyperbolic Volume 10.956 A-Polynomial See Data:K11n81/A-polynomial

[edit Notes for K11n81's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -6

[edit Notes for K11n81's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+3t^{3}-4t^{2}+4t-3+4t^{-1}-4t^{-2}+3t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-5z^{6}-6z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{2}-t+1\right\}$
Determinant and Signature	{ 27, 6 }
Jones polynomial	$2q^{9}-3q^{8}+3q^{7}-5q^{6}+4q^{5}-4q^{4}+4q^{3}-q^{2}+q$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-18z^{4}a^{-6}+6z^{4}a^{-8}+12z^{2}a^{-4}-23z^{2}a^{-6}+11z^{2}a^{-8}-z^{2}a^{-10}+8a^{-4}-13a^{-6}+7a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+5z^{8}a^{-6}+4z^{8}a^{-8}-4z^{7}a^{-5}+2z^{7}a^{-7}+6z^{7}a^{-9}-7z^{6}a^{-4}-28z^{6}a^{-6}-18z^{6}a^{-8}+3z^{6}a^{-10}-30z^{5}a^{-7}-30z^{5}a^{-9}+18z^{4}a^{-4}+50z^{4}a^{-6}+20z^{4}a^{-8}-12z^{4}a^{-10}+13z^{3}a^{-5}+53z^{3}a^{-7}+41z^{3}a^{-9}+z^{3}a^{-11}-20z^{2}a^{-4}-38z^{2}a^{-6}-10z^{2}a^{-8}+8z^{2}a^{-10}-11za^{-5}-26za^{-7}-18za^{-9}-3za^{-11}+8a^{-4}+13a^{-6}+7a^{-8}+a^{-10}$
The A2 invariant	$q^{-4}+q^{-6}+3q^{-8}+3q^{-10}+2q^{-12}-4q^{-16}-3q^{-18}-5q^{-20}+q^{-24}+2q^{-26}+2q^{-28}+q^{-32}-q^{-34}$
The G2 invariant	Data:K11n81/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n81 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["K11n81"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+3t^{3}-4t^{2}+4t-3+4t^{-1}-4t^{-2}+3t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-5z^{6}-6z^{4}-z^{2}+1$

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]=

$\left\{t^{2}-t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 27, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$2q^{9}-3q^{8}+3q^{7}-5q^{6}+4q^{5}-4q^{4}+4q^{3}-q^{2}+q$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-18z^{4}a^{-6}+6z^{4}a^{-8}+12z^{2}a^{-4}-23z^{2}a^{-6}+11z^{2}a^{-8}-z^{2}a^{-10}+8a^{-4}-13a^{-6}+7a^{-8}-a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+5z^{8}a^{-6}+4z^{8}a^{-8}-4z^{7}a^{-5}+2z^{7}a^{-7}+6z^{7}a^{-9}-7z^{6}a^{-4}-28z^{6}a^{-6}-18z^{6}a^{-8}+3z^{6}a^{-10}-30z^{5}a^{-7}-30z^{5}a^{-9}+18z^{4}a^{-4}+50z^{4}a^{-6}+20z^{4}a^{-8}-12z^{4}a^{-10}+13z^{3}a^{-5}+53z^{3}a^{-7}+41z^{3}a^{-9}+z^{3}a^{-11}-20z^{2}a^{-4}-38z^{2}a^{-6}-10z^{2}a^{-8}+8z^{2}a^{-10}-11za^{-5}-26za^{-7}-18za^{-9}-3za^{-11}+8a^{-4}+13a^{-6}+7a^{-8}+a^{-10}$

"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["K11n81"];

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]=

{ $-t^{4}+3t^{3}-4t^{2}+4t-3+4t^{-1}-4t^{-2}+3t^{-3}-t^{-4}$, $2q^{9}-3q^{8}+3q^{7}-5q^{6}+4q^{5}-4q^{4}+4q^{3}-q^{2}+q$ }

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, -6)

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 $-4$ $-48$ $8$ $-{\frac {302}{3}}$ ${\frac {158}{3}}$ $192$ $416$ $256$ $272$ $-{\frac {32}{3}}$ $1152$ ${\frac {1208}{3}}$ $-{\frac {632}{3}}$ ${\frac {123809}{30}}$ ${\frac {7502}{15}}$ ${\frac {83218}{45}}$ ${\frac {6559}{18}}$ $-{\frac {991}{30}}$

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=$6 is the signature of K11n81. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 χ 19                 2 2 17               1   -1 15             3 3   0 13           3 1     -2 11         1 3 1     -1 9       3 3         0 7     1 1           0 5   1 4             3 3                   0 1 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $i=7$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{4}$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }$ $r=6$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$

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.

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