K11a72

K11a71

K11a73

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a72 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a72/ThurstonBennequinNumber Hyperbolic Volume 17.173 A-Polynomial See Data:K11a72/A-polynomial

[edit Notes for K11a72's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[2,4]$ Rasmussen s-Invariant 0

[edit Notes for K11a72's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+18t^{2}-32t+39-32t^{-1}+18t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}+2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 153, 0 }
Jones polynomial	$q^{6}-5q^{5}+10q^{4}-16q^{3}+22q^{2}-24q+25-21q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-6z^{4}a^{-2}+z^{4}a^{-4}+10z^{4}-3a^{2}z^{2}-4z^{2}a^{-2}+z^{2}a^{-4}+8z^{2}-a^{2}+a^{-2}-a^{-4}+2$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+7az^{9}+14z^{9}a^{-1}+7z^{9}a^{-3}+10a^{2}z^{8}+19z^{8}a^{-2}+9z^{8}a^{-4}+20z^{8}+8a^{3}z^{7}+az^{7}-16z^{7}a^{-1}-4z^{7}a^{-3}+5z^{7}a^{-5}+4a^{4}z^{6}-15a^{2}z^{6}-55z^{6}a^{-2}-20z^{6}a^{-4}+z^{6}a^{-6}-53z^{6}+a^{5}z^{5}-12a^{3}z^{5}-20az^{5}-15z^{5}a^{-1}-18z^{5}a^{-3}-10z^{5}a^{-5}-5a^{4}z^{4}+11a^{2}z^{4}+44z^{4}a^{-2}+12z^{4}a^{-4}-z^{4}a^{-6}+47z^{4}-a^{5}z^{3}+7a^{3}z^{3}+20az^{3}+22z^{3}a^{-1}+14z^{3}a^{-3}+4z^{3}a^{-5}+a^{4}z^{2}-5a^{2}z^{2}-12z^{2}a^{-2}-2z^{2}a^{-4}-16z^{2}-2a^{3}z-5az-5za^{-1}-za^{-3}+za^{-5}+a^{2}-a^{-2}-a^{-4}+2$
The A2 invariant	$-q^{14}+2q^{12}-3q^{10}+2q^{8}+q^{6}-4q^{4}+5q^{2}-4+4q^{-2}+q^{-4}+5q^{-8}-4q^{-10}+q^{-12}-q^{-14}-2q^{-16}+q^{-18}$
The G2 invariant	$q^{80}-3q^{78}+7q^{76}-13q^{74}+16q^{72}-16q^{70}+7q^{68}+16q^{66}-45q^{64}+83q^{62}-114q^{60}+115q^{58}-80q^{56}-9q^{54}+139q^{52}-278q^{50}+389q^{48}-410q^{46}+295q^{44}-41q^{42}-306q^{40}+643q^{38}-839q^{36}+789q^{34}-473q^{32}-53q^{30}+605q^{28}-976q^{26}+1019q^{24}-679q^{22}+96q^{20}+491q^{18}-837q^{16}+777q^{14}-340q^{12}-279q^{10}+797q^{8}-952q^{6}+644q^{4}+28q^{2}-796+1345q^{-2}-1425q^{-4}+976q^{-6}-156q^{-8}-748q^{-10}+1412q^{-12}-1590q^{-14}+1240q^{-16}-491q^{-18}-358q^{-20}+997q^{-22}-1192q^{-24}+906q^{-26}-281q^{-28}-387q^{-30}+813q^{-32}-819q^{-34}+412q^{-36}+228q^{-38}-798q^{-40}+1054q^{-42}-878q^{-44}+334q^{-46}+339q^{-48}-892q^{-50}+1112q^{-52}-949q^{-54}+500q^{-56}+44q^{-58}-492q^{-60}+703q^{-62}-662q^{-64}+443q^{-66}-155q^{-68}-92q^{-70}+226q^{-72}-253q^{-74}+196q^{-76}-106q^{-78}+32q^{-80}+21q^{-82}-39q^{-84}+35q^{-86}-24q^{-88}+11q^{-90}-4q^{-92}+q^{-94}$

Further Quantum Invariants

K11a72 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["K11a72"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+18t^{2}-32t+39-32t^{-1}+18t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 153, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-5q^{5}+10q^{4}-16q^{3}+22q^{2}-24q+25-21q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+7az^{9}+14z^{9}a^{-1}+7z^{9}a^{-3}+10a^{2}z^{8}+19z^{8}a^{-2}+9z^{8}a^{-4}+20z^{8}+8a^{3}z^{7}+az^{7}-16z^{7}a^{-1}-4z^{7}a^{-3}+5z^{7}a^{-5}+4a^{4}z^{6}-15a^{2}z^{6}-55z^{6}a^{-2}-20z^{6}a^{-4}+z^{6}a^{-6}-53z^{6}+a^{5}z^{5}-12a^{3}z^{5}-20az^{5}-15z^{5}a^{-1}-18z^{5}a^{-3}-10z^{5}a^{-5}-5a^{4}z^{4}+11a^{2}z^{4}+44z^{4}a^{-2}+12z^{4}a^{-4}-z^{4}a^{-6}+47z^{4}-a^{5}z^{3}+7a^{3}z^{3}+20az^{3}+22z^{3}a^{-1}+14z^{3}a^{-3}+4z^{3}a^{-5}+a^{4}z^{2}-5a^{2}z^{2}-12z^{2}a^{-2}-2z^{2}a^{-4}-16z^{2}-2a^{3}z-5az-5za^{-1}-za^{-3}+za^{-5}+a^{2}-a^{-2}-a^{-4}+2$

"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["K11a72"];

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]=

{ $t^{4}-6t^{3}+18t^{2}-32t+39-32t^{-1}+18t^{-2}-6t^{-3}+t^{-4}$, $q^{6}-5q^{5}+10q^{4}-16q^{3}+22q^{2}-24q+25-21q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$ }

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:

(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 $8$ $8$ $32$ ${\frac {124}{3}}$ $-{\frac {4}{3}}$ $64$ ${\frac {368}{3}}$ ${\frac {128}{3}}$ $8$ ${\frac {256}{3}}$ $32$ ${\frac {992}{3}}$ $-{\frac {32}{3}}$ ${\frac {6751}{15}}$ $-{\frac {164}{15}}$ ${\frac {7684}{45}}$ $-{\frac {31}{9}}$ ${\frac {271}{15}}$

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 K11a72. 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                   6 1   5 7                 10 4     -6 5               12 6       6 3             12 10         -2 1           13 12           1 -1         9 13             4 -3       6 12               -6 -5     3 9                 6 -7   1 6                   -5 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a71 K11a73