K11n75

K11n74

K11n76

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n75 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,14,6,15 X2837 X9,20,10,21 X11,16,12,17 X13,6,14,7 X15,18,16,19 X17,12,18,13 X19,22,20,1 X21,10,22,11
Gauss code	1, -4, 2, -1, -3, 7, 4, -2, -5, 11, -6, 9, -7, 3, -8, 6, -9, 8, -10, 5, -11, 10
Dowker-Thistlethwaite code	4 8 -14 2 -20 -16 -6 -18 -12 -22 -10

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 4 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n75/ThurstonBennequinNumber Hyperbolic Volume 13.5931 A-Polynomial See Data:K11n75/A-polynomial

[edit Notes for K11n75's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $1$ Rasmussen s-Invariant 2

[edit Notes for K11n75's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-7t^{2}+14t-17+14t^{-1}-7t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+5z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{2}-t+1\right\}$
Determinant and Signature	{ 63, -2 }
Jones polynomial	$-2+5q^{-1}-7q^{-2}+11q^{-3}-10q^{-4}+10q^{-5}-9q^{-6}+5q^{-7}-3q^{-8}+q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{8}+2a^{8}-3z^{4}a^{6}-10z^{2}a^{6}-9a^{6}+2z^{6}a^{4}+10z^{4}a^{4}+18z^{2}a^{4}+11a^{4}-2z^{4}a^{2}-5z^{2}a^{2}-3a^{2}$
Kauffman polynomial (db, data sources)	$z^{6}a^{10}-3z^{4}a^{10}+2z^{2}a^{10}+3z^{7}a^{9}-10z^{5}a^{9}+10z^{3}a^{9}-4za^{9}+3z^{8}a^{8}-6z^{6}a^{8}-z^{4}a^{8}+z^{2}a^{8}+2a^{8}+z^{9}a^{7}+7z^{7}a^{7}-31z^{5}a^{7}+36z^{3}a^{7}-17za^{7}+7z^{8}a^{6}-18z^{6}a^{6}+17z^{4}a^{6}-15z^{2}a^{6}+9a^{6}+z^{9}a^{5}+8z^{7}a^{5}-31z^{5}a^{5}+42z^{3}a^{5}-21za^{5}+4z^{8}a^{4}-10z^{6}a^{4}+19z^{4}a^{4}-20z^{2}a^{4}+11a^{4}+4z^{7}a^{3}-10z^{5}a^{3}+19z^{3}a^{3}-11za^{3}+z^{6}a^{2}+4z^{4}a^{2}-6z^{2}a^{2}+3a^{2}+3z^{3}a-3za$
The A2 invariant	Data:K11n75/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n75/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n75 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["K11n75"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-7t^{2}+14t-17+14t^{-1}-7t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}+5z^{4}+4z^{2}+1$

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]=

$\left\{t^{2}-t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 63, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{8}+2a^{8}-3z^{4}a^{6}-10z^{2}a^{6}-9a^{6}+2z^{6}a^{4}+10z^{4}a^{4}+18z^{2}a^{4}+11a^{4}-2z^{4}a^{2}-5z^{2}a^{2}-3a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{6}a^{10}-3z^{4}a^{10}+2z^{2}a^{10}+3z^{7}a^{9}-10z^{5}a^{9}+10z^{3}a^{9}-4za^{9}+3z^{8}a^{8}-6z^{6}a^{8}-z^{4}a^{8}+z^{2}a^{8}+2a^{8}+z^{9}a^{7}+7z^{7}a^{7}-31z^{5}a^{7}+36z^{3}a^{7}-17za^{7}+7z^{8}a^{6}-18z^{6}a^{6}+17z^{4}a^{6}-15z^{2}a^{6}+9a^{6}+z^{9}a^{5}+8z^{7}a^{5}-31z^{5}a^{5}+42z^{3}a^{5}-21za^{5}+4z^{8}a^{4}-10z^{6}a^{4}+19z^{4}a^{4}-20z^{2}a^{4}+11a^{4}+4z^{7}a^{3}-10z^{5}a^{3}+19z^{3}a^{3}-11za^{3}+z^{6}a^{2}+4z^{4}a^{2}-6z^{2}a^{2}+3a^{2}+3z^{3}a-3za$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_65, 10_77, K11n71,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n71,}

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["K11n75"];

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]=

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

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]=

{10_65, 10_77, K11n71,}

In[6]:=

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

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

Out[6]=

{K11n71,}

Vassiliev invariants

V2 and V3:

(4, -5)

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 $16$ $-40$ $128$ ${\frac {536}{3}}$ ${\frac {64}{3}}$ $-640$ $-{\frac {2416}{3}}$ $-{\frac {352}{3}}$ $-104$ ${\frac {2048}{3}}$ $800$ ${\frac {8576}{3}}$ ${\frac {1024}{3}}$ ${\frac {58622}{15}}$ ${\frac {3632}{15}}$ ${\frac {53888}{45}}$ ${\frac {370}{9}}$ ${\frac {1982}{15}}$

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 $s=$-2 is the signature of K11n75. 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                   2 -2 -1                 3   3 -3               5 3   -2 -5             6 2     4 -7           4 5       1 -9         6 6         0 -11       3 4           1 -13     2 6             -4 -15   1 3               2 -17   2                 -2 -19 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n74 K11n76