K11a233

K11a232

K11a234

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a233 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X14,4,15,3 X10,5,11,6 X20,8,21,7 X22,9,1,10 X18,11,19,12 X2,14,3,13 X8,15,9,16 X6,18,7,17 X12,19,13,20 X16,22,17,21
Gauss code	1, -7, 2, -1, 3, -9, 4, -8, 5, -3, 6, -10, 7, -2, 8, -11, 9, -6, 10, -4, 11, -5
Dowker-Thistlethwaite code	4 14 10 20 22 18 2 8 6 12 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a233/ThurstonBennequinNumber Hyperbolic Volume 18.7296 A-Polynomial See Data:K11a233/A-polynomial

[edit Notes for K11a233'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 K11a233's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+19t^{2}-37t+47-37t^{-1}+19t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}+3z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 173, 0 }
Jones polynomial	$q^{6}-5q^{5}+11q^{4}-18q^{3}+25q^{2}-28q+28-24q^{-1}+18q^{-2}-10q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-6z^{4}a^{-2}+z^{4}a^{-4}+11z^{4}-4a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}+10z^{2}-a^{2}-a^{-2}+3$
Kauffman polynomial (db, data sources)	$3z^{10}a^{-2}+3z^{10}+10az^{9}+19z^{9}a^{-1}+9z^{9}a^{-3}+13a^{2}z^{8}+21z^{8}a^{-2}+10z^{8}a^{-4}+24z^{8}+9a^{3}z^{7}-4az^{7}-26z^{7}a^{-1}-8z^{7}a^{-3}+5z^{7}a^{-5}+4a^{4}z^{6}-21a^{2}z^{6}-62z^{6}a^{-2}-21z^{6}a^{-4}+z^{6}a^{-6}-65z^{6}+a^{5}z^{5}-12a^{3}z^{5}-16az^{5}-8z^{5}a^{-1}-14z^{5}a^{-3}-9z^{5}a^{-5}-4a^{4}z^{4}+18a^{2}z^{4}+48z^{4}a^{-2}+13z^{4}a^{-4}-z^{4}a^{-6}+56z^{4}-a^{5}z^{3}+8a^{3}z^{3}+17az^{3}+17z^{3}a^{-1}+13z^{3}a^{-3}+4z^{3}a^{-5}+a^{4}z^{2}-8a^{2}z^{2}-14z^{2}a^{-2}-2z^{2}a^{-4}-21z^{2}-2a^{3}z-5az-5za^{-1}-2za^{-3}+a^{2}+a^{-2}+3$
The A2 invariant	$-q^{14}+2q^{12}-4q^{10}+3q^{8}+2q^{6}-4q^{4}+6q^{2}-5+4q^{-2}-q^{-6}+5q^{-8}-5q^{-10}+2q^{-12}-2q^{-16}+q^{-18}$
The G2 invariant	$q^{80}-3q^{78}+7q^{76}-13q^{74}+17q^{72}-19q^{70}+12q^{68}+10q^{66}-43q^{64}+89q^{62}-134q^{60}+156q^{58}-137q^{56}+47q^{54}+124q^{52}-341q^{50}+547q^{48}-650q^{46}+539q^{44}-191q^{42}-364q^{40}+970q^{38}-1370q^{36}+1362q^{34}-847q^{32}-58q^{30}+1029q^{28}-1692q^{26}+1741q^{24}-1115q^{22}+53q^{20}+995q^{18}-1553q^{16}+1370q^{14}-502q^{12}-631q^{10}+1504q^{8}-1703q^{6}+1078q^{4}+119q^{2}-1412+2279q^{-2}-2327q^{-4}+1526q^{-6}-137q^{-8}-1329q^{-10}+2335q^{-12}-2541q^{-14}+1875q^{-16}-610q^{-18}-769q^{-20}+1750q^{-22}-1955q^{-24}+1364q^{-26}-237q^{-28}-899q^{-30}+1524q^{-32}-1392q^{-34}+552q^{-36}+590q^{-38}-1509q^{-40}+1818q^{-42}-1366q^{-44}+367q^{-46}+751q^{-48}-1552q^{-50}+1747q^{-52}-1340q^{-54}+554q^{-56}+263q^{-58}-838q^{-60}+1029q^{-62}-862q^{-64}+498q^{-66}-99q^{-68}-190q^{-70}+312q^{-72}-306q^{-74}+213q^{-76}-101q^{-78}+20q^{-80}+30q^{-82}-43q^{-84}+36q^{-86}-24q^{-88}+11q^{-90}-4q^{-92}+q^{-94}$

Further Quantum Invariants

K11a233 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["K11a233"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+19t^{2}-37t+47-37t^{-1}+19t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+2z^{6}+3z^{4}+z^{2}+1$

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]=

{ 173, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-5q^{5}+11q^{4}-18q^{3}+25q^{2}-28q+28-24q^{-1}+18q^{-2}-10q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-6z^{4}a^{-2}+z^{4}a^{-4}+11z^{4}-4a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}+10z^{2}-a^{2}-a^{-2}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3z^{10}a^{-2}+3z^{10}+10az^{9}+19z^{9}a^{-1}+9z^{9}a^{-3}+13a^{2}z^{8}+21z^{8}a^{-2}+10z^{8}a^{-4}+24z^{8}+9a^{3}z^{7}-4az^{7}-26z^{7}a^{-1}-8z^{7}a^{-3}+5z^{7}a^{-5}+4a^{4}z^{6}-21a^{2}z^{6}-62z^{6}a^{-2}-21z^{6}a^{-4}+z^{6}a^{-6}-65z^{6}+a^{5}z^{5}-12a^{3}z^{5}-16az^{5}-8z^{5}a^{-1}-14z^{5}a^{-3}-9z^{5}a^{-5}-4a^{4}z^{4}+18a^{2}z^{4}+48z^{4}a^{-2}+13z^{4}a^{-4}-z^{4}a^{-6}+56z^{4}-a^{5}z^{3}+8a^{3}z^{3}+17az^{3}+17z^{3}a^{-1}+13z^{3}a^{-3}+4z^{3}a^{-5}+a^{4}z^{2}-8a^{2}z^{2}-14z^{2}a^{-2}-2z^{2}a^{-4}-21z^{2}-2a^{3}z-5az-5za^{-1}-2za^{-3}+a^{2}+a^{-2}+3$

"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["K11a233"];

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]=

{ $t^{4}-6t^{3}+19t^{2}-37t+47-37t^{-1}+19t^{-2}-6t^{-3}+t^{-4}$, $q^{6}-5q^{5}+11q^{4}-18q^{3}+25q^{2}-28q+28-24q^{-1}+18q^{-2}-10q^{-3}+4q^{-4}-q^{-5}$ }

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]=

{}

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 $4$ $0$ $8$ $-{\frac {34}{3}}$ $-{\frac {62}{3}}$ $0$ $0$ $32$ $-32$ ${\frac {32}{3}}$ $0$ $-{\frac {136}{3}}$ $-{\frac {248}{3}}$ $-{\frac {1649}{30}}$ ${\frac {86}{5}}$ $-{\frac {3778}{45}}$ $-{\frac {271}{18}}$ $-{\frac {689}{30}}$

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 K11a233. 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                   7 1   6 7                 11 4     -7 5               14 7       7 3             14 11         -3 1           14 14           0 -1         11 15             4 -3       7 13               -6 -5     3 11                 8 -7   1 7                   -6 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a232 K11a234