K11n87

K11n86

K11n88

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n87 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X18,9,19,10 X2,11,3,12 X20,14,21,13 X8,15,9,16 X17,22,18,1 X6,19,7,20 X21,16,22,17
Gauss code	1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -4, 8, 11, -9, -5, 10, -7, -11, 9
Dowker-Thistlethwaite code	4 10 12 14 18 2 20 8 -22 6 -16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n87/ThurstonBennequinNumber Hyperbolic Volume 12.3434 A-Polynomial See Data:K11n87/A-polynomial

[edit Notes for K11n87's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for K11n87's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-5t^{2}+12t-15+12t^{-1}-5t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 51, -2 }
Jones polynomial	$-1+4q^{-1}-5q^{-2}+8q^{-3}-9q^{-4}+8q^{-5}-7q^{-6}+5q^{-7}-3q^{-8}+q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{8}+a^{8}-2z^{4}a^{6}-5z^{2}a^{6}-3a^{6}+z^{6}a^{4}+4z^{4}a^{4}+6z^{2}a^{4}+2a^{4}-z^{4}a^{2}-z^{2}a^{2}+a^{2}$
Kauffman polynomial (db, data sources)	$z^{6}a^{10}-3z^{4}a^{10}+2z^{2}a^{10}+3z^{7}a^{9}-10z^{5}a^{9}+8z^{3}a^{9}-2za^{9}+3z^{8}a^{8}-8z^{6}a^{8}+3z^{4}a^{8}+a^{8}+z^{9}a^{7}+3z^{7}a^{7}-16z^{5}a^{7}+14z^{3}a^{7}-4za^{7}+5z^{8}a^{6}-14z^{6}a^{6}+14z^{4}a^{6}-9z^{2}a^{6}+3a^{6}+z^{9}a^{5}+z^{7}a^{5}-5z^{5}a^{5}+5z^{3}a^{5}-za^{5}+2z^{8}a^{4}-5z^{6}a^{4}+12z^{4}a^{4}-10z^{2}a^{4}+2a^{4}+z^{7}a^{3}+z^{5}a^{3}+za^{3}+4z^{4}a^{2}-3z^{2}a^{2}-a^{2}+z^{3}a$
The A2 invariant	$q^{28}-q^{24}+q^{22}-2q^{20}-2q^{14}+q^{12}-q^{10}+3q^{8}+2q^{6}+2q^{2}-1$
The G2 invariant	Data:K11n87/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n87 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["K11n87"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-5t^{2}+12t-15+12t^{-1}-5t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 51, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{6}a^{10}-3z^{4}a^{10}+2z^{2}a^{10}+3z^{7}a^{9}-10z^{5}a^{9}+8z^{3}a^{9}-2za^{9}+3z^{8}a^{8}-8z^{6}a^{8}+3z^{4}a^{8}+a^{8}+z^{9}a^{7}+3z^{7}a^{7}-16z^{5}a^{7}+14z^{3}a^{7}-4za^{7}+5z^{8}a^{6}-14z^{6}a^{6}+14z^{4}a^{6}-9z^{2}a^{6}+3a^{6}+z^{9}a^{5}+z^{7}a^{5}-5z^{5}a^{5}+5z^{3}a^{5}-za^{5}+2z^{8}a^{4}-5z^{6}a^{4}+12z^{4}a^{4}-10z^{2}a^{4}+2a^{4}+z^{7}a^{3}+z^{5}a^{3}+za^{3}+4z^{4}a^{2}-3z^{2}a^{2}-a^{2}+z^{3}a$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_28, 9_29, 10_163,}

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["K11n87"];

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]=

{ $t^{3}-5t^{2}+12t-15+12t^{-1}-5t^{-2}+t^{-3}$, $-1+4q^{-1}-5q^{-2}+8q^{-3}-9q^{-4}+8q^{-5}-7q^{-6}+5q^{-7}-3q^{-8}+q^{-9}$ }

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]=

{9_28, 9_29, 10_163,}

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 $4$ $0$ $8$ $-{\frac {82}{3}}$ $-{\frac {14}{3}}$ $0$ $128$ $0$ $32$ ${\frac {32}{3}}$ $0$ $-{\frac {328}{3}}$ $-{\frac {56}{3}}$ $-{\frac {12449}{30}}$ ${\frac {326}{5}}$ $-{\frac {7618}{45}}$ $-{\frac {511}{18}}$ $-{\frac {449}{30}}$

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 $s=$-2 is the signature of K11n87. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n86 K11n88