K11a71

K11a70

K11a72

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a71 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 X18,13,19,14 X20,16,21,15 X6,18,7,17 X8,19,9,20 X16,22,17,21
Gauss code	1, -5, 2, -1, 3, -9, 4, -10, 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 6 8 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:K11a71/ThurstonBennequinNumber Hyperbolic Volume 17.3873 A-Polynomial See Data:K11a71/A-polynomial

[edit Notes for K11a71's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -2

[edit Notes for K11a71's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-18t^{2}+34t-41+34t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}-2z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 159, 2 }
Jones polynomial	$-q^{8}+5q^{7}-11q^{6}+17q^{5}-23q^{4}+26q^{3}-25q^{2}+22q-15+9q^{-1}-4q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-10z^{4}a^{-2}+6z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-7z^{2}a^{-2}+5z^{2}a^{-4}-z^{2}a^{-6}+3z^{2}+1$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}a^{-4}+6z^{9}a^{-1}+14z^{9}a^{-3}+8z^{9}a^{-5}+17z^{8}a^{-2}+23z^{8}a^{-4}+13z^{8}a^{-6}+7z^{8}+4az^{7}-3z^{7}a^{-1}-13z^{7}a^{-3}+5z^{7}a^{-5}+11z^{7}a^{-7}+a^{2}z^{6}-47z^{6}a^{-2}-53z^{6}a^{-4}-17z^{6}a^{-6}+5z^{6}a^{-8}-15z^{6}-9az^{5}-14z^{5}a^{-1}-18z^{5}a^{-3}-29z^{5}a^{-5}-15z^{5}a^{-7}+z^{5}a^{-9}-2a^{2}z^{4}+39z^{4}a^{-2}+35z^{4}a^{-4}+5z^{4}a^{-6}-4z^{4}a^{-8}+11z^{4}+6az^{3}+14z^{3}a^{-1}+20z^{3}a^{-3}+17z^{3}a^{-5}+5z^{3}a^{-7}+a^{2}z^{2}-12z^{2}a^{-2}-8z^{2}a^{-4}-z^{2}a^{-6}-4z^{2}-az-3za^{-1}-3za^{-3}-za^{-5}+1$
The A2 invariant	$q^{8}-2q^{6}+3q^{4}-2q^{2}-1+5q^{-2}-4q^{-4}+6q^{-6}-2q^{-8}+q^{-12}-5q^{-14}+4q^{-16}-2q^{-18}+2q^{-22}-q^{-24}$
The G2 invariant	$q^{46}-3q^{44}+8q^{42}-16q^{40}+22q^{38}-25q^{36}+14q^{34}+18q^{32}-65q^{30}+127q^{28}-176q^{26}+178q^{24}-110q^{22}-51q^{20}+277q^{18}-495q^{16}+619q^{14}-551q^{12}+254q^{10}+221q^{8}-733q^{6}+1094q^{4}-1121q^{2}+759-102q^{-2}-637q^{-4}+1158q^{-6}-1252q^{-8}+878q^{-10}-176q^{-12}-542q^{-14}+971q^{-16}-920q^{-18}+406q^{-20}+341q^{-22}-974q^{-24}+1199q^{-26}-873q^{-28}+94q^{-30}+840q^{-32}-1553q^{-34}+1765q^{-36}-1353q^{-38}+450q^{-40}+627q^{-42}-1496q^{-44}+1839q^{-46}-1553q^{-48}+761q^{-50}+212q^{-52}-989q^{-54}+1279q^{-56}-1014q^{-58}+349q^{-60}+402q^{-62}-898q^{-64}+913q^{-66}-467q^{-68}-244q^{-70}+900q^{-72}-1206q^{-74}+1053q^{-76}-495q^{-78}-230q^{-80}+847q^{-82}-1157q^{-84}+1079q^{-86}-685q^{-88}+156q^{-90}+320q^{-92}-610q^{-94}+664q^{-96}-522q^{-98}+289q^{-100}-49q^{-102}-126q^{-104}+202q^{-106}-205q^{-108}+147q^{-110}-76q^{-112}+22q^{-114}+16q^{-116}-28q^{-118}+27q^{-120}-20q^{-122}+10q^{-124}-4q^{-126}+q^{-128}$

Further Quantum Invariants

K11a71 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["K11a71"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-18t^{2}+34t-41+34t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 159, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+5q^{7}-11q^{6}+17q^{5}-23q^{4}+26q^{3}-25q^{2}+22q-15+9q^{-1}-4q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-10z^{4}a^{-2}+6z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-7z^{2}a^{-2}+5z^{2}a^{-4}-z^{2}a^{-6}+3z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}a^{-4}+6z^{9}a^{-1}+14z^{9}a^{-3}+8z^{9}a^{-5}+17z^{8}a^{-2}+23z^{8}a^{-4}+13z^{8}a^{-6}+7z^{8}+4az^{7}-3z^{7}a^{-1}-13z^{7}a^{-3}+5z^{7}a^{-5}+11z^{7}a^{-7}+a^{2}z^{6}-47z^{6}a^{-2}-53z^{6}a^{-4}-17z^{6}a^{-6}+5z^{6}a^{-8}-15z^{6}-9az^{5}-14z^{5}a^{-1}-18z^{5}a^{-3}-29z^{5}a^{-5}-15z^{5}a^{-7}+z^{5}a^{-9}-2a^{2}z^{4}+39z^{4}a^{-2}+35z^{4}a^{-4}+5z^{4}a^{-6}-4z^{4}a^{-8}+11z^{4}+6az^{3}+14z^{3}a^{-1}+20z^{3}a^{-3}+17z^{3}a^{-5}+5z^{3}a^{-7}+a^{2}z^{2}-12z^{2}a^{-2}-8z^{2}a^{-4}-z^{2}a^{-6}-4z^{2}-az-3za^{-1}-3za^{-3}-za^{-5}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a248,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a248,}

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["K11a71"];

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]=

{ $-t^{4}+6t^{3}-18t^{2}+34t-41+34t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$, $-q^{8}+5q^{7}-11q^{6}+17q^{5}-23q^{4}+26q^{3}-25q^{2}+22q-15+9q^{-1}-4q^{-2}+q^{-3}$ }

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]=

{K11a248,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11a248,}

Vassiliev invariants

V2 and V3:

(0, 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 $0$ $0$ $0$ $16$ $16$ $0$ $32$ $0$ $32$ $0$ $0$ $0$ $0$ $40$ ${\frac {176}{3}}$ $-{\frac {160}{3}}$ ${\frac {104}{3}}$ $-40$

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=$2 is the signature of K11a71. 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 17                       1 -1 15                     4   4 13                   7 1   -6 11                 10 4     6 9               13 7       -6 7             13 10         3 5           12 13           1 3         10 13             -3 1       6 13               7 -1     3 9                 -6 -3   1 6                   5 -5   3                     -3 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a70 K11a72