K11a40

K11a39

K11a41

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a40 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X18,10,19,9 X20,12,21,11 X16,13,17,14 X6,15,7,16 X22,18,1,17 X10,20,11,19 X12,22,13,21
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -10, 6, -11, 7, -3, 8, -7, 9, -5, 10, -6, 11, -9
Dowker-Thistlethwaite code	4 8 14 2 18 20 16 6 22 10 12

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:K11a40/ThurstonBennequinNumber Hyperbolic Volume 12.8867 A-Polynomial See Data:K11a40/A-polynomial

[edit Notes for K11a40'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 K11a40's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+12t^{2}-17t+19-17t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 89, 4 }
Jones polynomial	$q^{10}-3q^{9}+6q^{8}-10q^{7}+12q^{6}-14q^{5}+14q^{4}-11q^{3}+9q^{2}-5q+3-q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-z^{6}a^{-2}+6z^{6}a^{-4}-2z^{6}a^{-6}-4z^{4}a^{-2}+14z^{4}a^{-4}-9z^{4}a^{-6}+z^{4}a^{-8}-4z^{2}a^{-2}+16z^{2}a^{-4}-13z^{2}a^{-6}+3z^{2}a^{-8}-a^{-2}+7a^{-4}-7a^{-6}+2a^{-8}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-4}+z^{10}a^{-6}+3z^{9}a^{-3}+6z^{9}a^{-5}+3z^{9}a^{-7}+3z^{8}a^{-2}+5z^{8}a^{-4}+7z^{8}a^{-6}+5z^{8}a^{-8}+z^{7}a^{-1}-9z^{7}a^{-3}-14z^{7}a^{-5}+2z^{7}a^{-7}+6z^{7}a^{-9}-13z^{6}a^{-2}-30z^{6}a^{-4}-26z^{6}a^{-6}-4z^{6}a^{-8}+5z^{6}a^{-10}-4z^{5}a^{-1}+2z^{5}a^{-3}-16z^{5}a^{-7}-7z^{5}a^{-9}+3z^{5}a^{-11}+17z^{4}a^{-2}+42z^{4}a^{-4}+30z^{4}a^{-6}-z^{4}a^{-8}-5z^{4}a^{-10}+z^{4}a^{-12}+4z^{3}a^{-1}+8z^{3}a^{-3}+14z^{3}a^{-5}+17z^{3}a^{-7}+4z^{3}a^{-9}-3z^{3}a^{-11}-8z^{2}a^{-2}-25z^{2}a^{-4}-21z^{2}a^{-6}-z^{2}a^{-8}+2z^{2}a^{-10}-z^{2}a^{-12}-za^{-1}-3za^{-3}-8za^{-5}-8za^{-7}-za^{-9}+za^{-11}+a^{-2}+7a^{-4}+7a^{-6}+2a^{-8}$
The A2 invariant	$-q^{2}+1-q^{-2}+q^{-4}+2q^{-6}+5q^{-10}-q^{-12}+2q^{-14}-q^{-16}-3q^{-18}-3q^{-22}+q^{-24}+q^{-30}$
The G2 invariant	$q^{12}-2q^{10}+5q^{8}-9q^{6}+10q^{4}-11q^{2}+3+14q^{-2}-36q^{-4}+57q^{-6}-65q^{-8}+46q^{-10}-4q^{-12}-58q^{-14}+118q^{-16}-146q^{-18}+129q^{-20}-62q^{-22}-37q^{-24}+130q^{-26}-180q^{-28}+171q^{-30}-98q^{-32}+q^{-34}+95q^{-36}-139q^{-38}+127q^{-40}-58q^{-42}-22q^{-44}+94q^{-46}-112q^{-48}+76q^{-50}+5q^{-52}-92q^{-54}+161q^{-56}-166q^{-58}+111q^{-60}-6q^{-62}-115q^{-64}+201q^{-66}-231q^{-68}+181q^{-70}-73q^{-72}-59q^{-74}+157q^{-76}-198q^{-78}+162q^{-80}-78q^{-82}-24q^{-84}+87q^{-86}-106q^{-88}+69q^{-90}-6q^{-92}-55q^{-94}+86q^{-96}-73q^{-98}+23q^{-100}+37q^{-102}-92q^{-104}+116q^{-106}-103q^{-108}+64q^{-110}-6q^{-112}-52q^{-114}+97q^{-116}-115q^{-118}+106q^{-120}-67q^{-122}+20q^{-124}+27q^{-126}-63q^{-128}+77q^{-130}-70q^{-132}+51q^{-134}-20q^{-136}-6q^{-138}+23q^{-140}-31q^{-142}+28q^{-144}-20q^{-146}+11q^{-148}-2q^{-150}-4q^{-152}+5q^{-154}-6q^{-156}+4q^{-158}-2q^{-160}+q^{-162}$

Further Quantum Invariants

K11a40 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["K11a40"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+12t^{2}-17t+19-17t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 89, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{10}-3q^{9}+6q^{8}-10q^{7}+12q^{6}-14q^{5}+14q^{4}-11q^{3}+9q^{2}-5q+3-q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-4}-z^{6}a^{-2}+6z^{6}a^{-4}-2z^{6}a^{-6}-4z^{4}a^{-2}+14z^{4}a^{-4}-9z^{4}a^{-6}+z^{4}a^{-8}-4z^{2}a^{-2}+16z^{2}a^{-4}-13z^{2}a^{-6}+3z^{2}a^{-8}-a^{-2}+7a^{-4}-7a^{-6}+2a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-4}+z^{10}a^{-6}+3z^{9}a^{-3}+6z^{9}a^{-5}+3z^{9}a^{-7}+3z^{8}a^{-2}+5z^{8}a^{-4}+7z^{8}a^{-6}+5z^{8}a^{-8}+z^{7}a^{-1}-9z^{7}a^{-3}-14z^{7}a^{-5}+2z^{7}a^{-7}+6z^{7}a^{-9}-13z^{6}a^{-2}-30z^{6}a^{-4}-26z^{6}a^{-6}-4z^{6}a^{-8}+5z^{6}a^{-10}-4z^{5}a^{-1}+2z^{5}a^{-3}-16z^{5}a^{-7}-7z^{5}a^{-9}+3z^{5}a^{-11}+17z^{4}a^{-2}+42z^{4}a^{-4}+30z^{4}a^{-6}-z^{4}a^{-8}-5z^{4}a^{-10}+z^{4}a^{-12}+4z^{3}a^{-1}+8z^{3}a^{-3}+14z^{3}a^{-5}+17z^{3}a^{-7}+4z^{3}a^{-9}-3z^{3}a^{-11}-8z^{2}a^{-2}-25z^{2}a^{-4}-21z^{2}a^{-6}-z^{2}a^{-8}+2z^{2}a^{-10}-z^{2}a^{-12}-za^{-1}-3za^{-3}-8za^{-5}-8za^{-7}-za^{-9}+za^{-11}+a^{-2}+7a^{-4}+7a^{-6}+2a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a330,}

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["K11a40"];

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]=

{ $t^{4}-5t^{3}+12t^{2}-17t+19-17t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$, $q^{10}-3q^{9}+6q^{8}-10q^{7}+12q^{6}-14q^{5}+14q^{4}-11q^{3}+9q^{2}-5q+3-q^{-1}$ }

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]=

{K11a330,}

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 $8$ $8$ $32$ $-{\frac {20}{3}}$ $-{\frac {4}{3}}$ $64$ $-{\frac {208}{3}}$ ${\frac {32}{3}}$ $8$ ${\frac {256}{3}}$ $32$ $-{\frac {160}{3}}$ $-{\frac {32}{3}}$ $-{\frac {5129}{15}}$ $-{\frac {2564}{15}}$ ${\frac {11284}{45}}$ $-{\frac {103}{9}}$ ${\frac {631}{15}}$

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 K11a40. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     2   -2 17                   4 1   3 15                 6 2     -4 13               6 4       2 11             8 6         -2 9           6 6           0 7         5 8             3 5       4 6               -2 3     2 6                 4 1   1 3                   -2 -1   2                     2 -3 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a39 K11a41