K11a270

K11a269

K11a271

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a270 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a270/ThurstonBennequinNumber Hyperbolic Volume 17.0324 A-Polynomial See Data:K11a270/A-polynomial

[edit Notes for K11a270's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11a270's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+12t^{2}-32t+45-32t^{-1}+12t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 137, 0 }
Jones polynomial	$q^{6}-4q^{5}+9q^{4}-14q^{3}+19q^{2}-22q+22-19q^{-1}+14q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+2a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}-z^{4}-a^{4}z^{2}+2a^{2}z^{2}-3z^{2}a^{-2}+z^{2}a^{-4}-z^{2}+a^{2}-a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$3z^{10}a^{-2}+3z^{10}+7az^{9}+15z^{9}a^{-1}+8z^{9}a^{-3}+8a^{2}z^{8}+7z^{8}a^{-2}+8z^{8}a^{-4}+7z^{8}+7a^{3}z^{7}-8az^{7}-39z^{7}a^{-1}-20z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-9a^{2}z^{6}-35z^{6}a^{-2}-22z^{6}a^{-4}+z^{6}a^{-6}-25z^{6}+a^{5}z^{5}-10a^{3}z^{5}+3az^{5}+40z^{5}a^{-1}+17z^{5}a^{-3}-9z^{5}a^{-5}-6a^{4}z^{4}-a^{2}z^{4}+39z^{4}a^{-2}+18z^{4}a^{-4}-2z^{4}a^{-6}+24z^{4}-a^{5}z^{3}+3a^{3}z^{3}-az^{3}-16z^{3}a^{-1}-8z^{3}a^{-3}+3z^{3}a^{-5}+2a^{4}z^{2}+3a^{2}z^{2}-14z^{2}a^{-2}-7z^{2}a^{-4}-6z^{2}+2za^{-1}+2za^{-3}-a^{2}+a^{-2}+a^{-4}$
The A2 invariant	Data:K11a270/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a270/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a270 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["K11a270"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+12t^{2}-32t+45-32t^{-1}+12t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{6}-2z^{2}+1$

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]=

{ 137, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-4q^{5}+9q^{4}-14q^{3}+19q^{2}-22q+22-19q^{-1}+14q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3z^{10}a^{-2}+3z^{10}+7az^{9}+15z^{9}a^{-1}+8z^{9}a^{-3}+8a^{2}z^{8}+7z^{8}a^{-2}+8z^{8}a^{-4}+7z^{8}+7a^{3}z^{7}-8az^{7}-39z^{7}a^{-1}-20z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-9a^{2}z^{6}-35z^{6}a^{-2}-22z^{6}a^{-4}+z^{6}a^{-6}-25z^{6}+a^{5}z^{5}-10a^{3}z^{5}+3az^{5}+40z^{5}a^{-1}+17z^{5}a^{-3}-9z^{5}a^{-5}-6a^{4}z^{4}-a^{2}z^{4}+39z^{4}a^{-2}+18z^{4}a^{-4}-2z^{4}a^{-6}+24z^{4}-a^{5}z^{3}+3a^{3}z^{3}-az^{3}-16z^{3}a^{-1}-8z^{3}a^{-3}+3z^{3}a^{-5}+2a^{4}z^{2}+3a^{2}z^{2}-14z^{2}a^{-2}-7z^{2}a^{-4}-6z^{2}+2za^{-1}+2za^{-3}-a^{2}+a^{-2}+a^{-4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a80,}

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["K11a270"];

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]=

{ $-2t^{3}+12t^{2}-32t+45-32t^{-1}+12t^{-2}-2t^{-3}$, $q^{6}-4q^{5}+9q^{4}-14q^{3}+19q^{2}-22q+22-19q^{-1}+14q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$ }

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]=

{K11a80,}

Vassiliev invariants

V2 and V3:

(-2, -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 $-8$ $-8$ $32$ ${\frac {116}{3}}$ ${\frac {52}{3}}$ $64$ ${\frac {304}{3}}$ ${\frac {256}{3}}$ $-40$ $-{\frac {256}{3}}$ $32$ $-{\frac {928}{3}}$ $-{\frac {416}{3}}$ $-{\frac {2791}{15}}$ ${\frac {2324}{15}}$ $-{\frac {15844}{45}}$ ${\frac {631}{9}}$ $-{\frac {1351}{15}}$

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 K11a270. 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                     3   -3 9                   6 1   5 7                 8 3     -5 5               11 6       5 3             11 8         -3 1           11 11           0 -1         9 12             3 -3       5 10               -5 -5     3 9                 6 -7   1 5                   -4 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a269 K11a271