K11a74

K11a73

K11a75

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a74 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a74/ThurstonBennequinNumber Hyperbolic Volume 12.3052 A-Polynomial See Data:K11a74/A-polynomial

[edit Notes for K11a74's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a74's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+10t^{2}-13t+15-13t^{-1}+10t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 73, 4 }
Jones polynomial	$-q^{9}+3q^{8}-5q^{7}+8q^{6}-10q^{5}+11q^{4}-11q^{3}+9q^{2}-7q+5-2q^{-1}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-10z^{4}a^{-2}+13z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-15z^{2}a^{-2}+13z^{2}a^{-4}-4z^{2}a^{-6}+4z^{2}-7a^{-2}+5a^{-4}-a^{-6}+4$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+z^{8}a^{-2}+7z^{8}a^{-4}+7z^{8}a^{-6}+z^{8}-10z^{7}a^{-1}-23z^{7}a^{-3}-5z^{7}a^{-5}+8z^{7}a^{-7}-23z^{6}a^{-2}-40z^{6}a^{-4}-16z^{6}a^{-6}+7z^{6}a^{-8}-6z^{6}+15z^{5}a^{-1}+18z^{5}a^{-3}-18z^{5}a^{-5}-16z^{5}a^{-7}+5z^{5}a^{-9}+47z^{4}a^{-2}+51z^{4}a^{-4}+5z^{4}a^{-6}-9z^{4}a^{-8}+3z^{4}a^{-10}+13z^{4}-6z^{3}a^{-1}+8z^{3}a^{-3}+26z^{3}a^{-5}+8z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}-32z^{2}a^{-2}-23z^{2}a^{-4}+2z^{2}a^{-8}-z^{2}a^{-10}-12z^{2}-za^{-1}-7za^{-3}-9za^{-5}-3za^{-7}+7a^{-2}+5a^{-4}+a^{-6}+4$
The A2 invariant	$q^{6}+q^{4}+q^{2}+2-q^{-2}-2q^{-6}-2q^{-8}+q^{-10}-2q^{-12}+3q^{-14}+q^{-18}+q^{-20}-q^{-22}+q^{-24}-q^{-26}$
The G2 invariant	$q^{26}-q^{24}+5q^{22}-7q^{20}+10q^{18}-9q^{16}+2q^{14}+15q^{12}-31q^{10}+46q^{8}-42q^{6}+23q^{4}+15q^{2}-53+84q^{-2}-81q^{-4}+53q^{-6}-2q^{-8}-52q^{-10}+82q^{-12}-81q^{-14}+50q^{-16}-4q^{-18}-40q^{-20}+57q^{-22}-50q^{-24}+11q^{-26}+25q^{-28}-58q^{-30}+60q^{-32}-40q^{-34}-7q^{-36}+51q^{-38}-87q^{-40}+94q^{-42}-70q^{-44}+20q^{-46}+40q^{-48}-85q^{-50}+103q^{-52}-82q^{-54}+41q^{-56}+18q^{-58}-54q^{-60}+67q^{-62}-48q^{-64}+15q^{-66}+24q^{-68}-40q^{-70}+35q^{-72}-9q^{-74}-21q^{-76}+43q^{-78}-46q^{-80}+32q^{-82}-8q^{-84}-19q^{-86}+35q^{-88}-43q^{-90}+38q^{-92}-25q^{-94}+9q^{-96}+5q^{-98}-20q^{-100}+27q^{-102}-31q^{-104}+28q^{-106}-18q^{-108}+7q^{-110}+7q^{-112}-19q^{-114}+23q^{-116}-23q^{-118}+18q^{-120}-8q^{-122}+7q^{-126}-12q^{-128}+13q^{-130}-10q^{-132}+7q^{-134}-2q^{-136}-q^{-138}+2q^{-140}-4q^{-142}+3q^{-144}-2q^{-146}+q^{-148}$

Further Quantum Invariants

K11a74 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["K11a74"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+10t^{2}-13t+15-13t^{-1}+10t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 73, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{9}+3q^{8}-5q^{7}+8q^{6}-10q^{5}+11q^{4}-11q^{3}+9q^{2}-7q+5-2q^{-1}+q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-10z^{4}a^{-2}+13z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-15z^{2}a^{-2}+13z^{2}a^{-4}-4z^{2}a^{-6}+4z^{2}-7a^{-2}+5a^{-4}-a^{-6}+4$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+z^{8}a^{-2}+7z^{8}a^{-4}+7z^{8}a^{-6}+z^{8}-10z^{7}a^{-1}-23z^{7}a^{-3}-5z^{7}a^{-5}+8z^{7}a^{-7}-23z^{6}a^{-2}-40z^{6}a^{-4}-16z^{6}a^{-6}+7z^{6}a^{-8}-6z^{6}+15z^{5}a^{-1}+18z^{5}a^{-3}-18z^{5}a^{-5}-16z^{5}a^{-7}+5z^{5}a^{-9}+47z^{4}a^{-2}+51z^{4}a^{-4}+5z^{4}a^{-6}-9z^{4}a^{-8}+3z^{4}a^{-10}+13z^{4}-6z^{3}a^{-1}+8z^{3}a^{-3}+26z^{3}a^{-5}+8z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}-32z^{2}a^{-2}-23z^{2}a^{-4}+2z^{2}a^{-8}-z^{2}a^{-10}-12z^{2}-za^{-1}-7za^{-3}-9za^{-5}-3za^{-7}+7a^{-2}+5a^{-4}+a^{-6}+4$

"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["K11a74"];

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]=

{ $t^{4}-5t^{3}+10t^{2}-13t+15-13t^{-1}+10t^{-2}-5t^{-3}+t^{-4}$, $-q^{9}+3q^{8}-5q^{7}+8q^{6}-10q^{5}+11q^{4}-11q^{3}+9q^{2}-7q+5-2q^{-1}+q^{-2}$ }

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 $-8$ $-8$ $32$ ${\frac {164}{3}}$ ${\frac {52}{3}}$ $64$ ${\frac {208}{3}}$ $-{\frac {128}{3}}$ $24$ $-{\frac {256}{3}}$ $32$ $-{\frac {1312}{3}}$ $-{\frac {416}{3}}$ $-{\frac {8911}{15}}$ $-{\frac {5276}{15}}$ $-{\frac {5044}{45}}$ ${\frac {127}{9}}$ ${\frac {209}{15}}$

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=$4 is the signature of K11a74. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a73 K11a75