K11n78

K11n77

K11n79

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n78 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 4 Bridge index 4 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n78/ThurstonBennequinNumber Hyperbolic Volume 12.3125 A-Polynomial See Data:K11n78/A-polynomial

[edit Notes for K11n78's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n78's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-3t^{3}+6t^{2}-8t+9-8t^{-1}+6t^{-2}-3t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+5z^{6}+8z^{4}+5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{2}-t+1\right\}$
Determinant and Signature	{ 45, 4 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^8-4 q^7+5 q^6-7 q^5+8 q^4-6 q^3+7 q^2-4 q+2- q^{-1} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +19 z^4 a^{-4} -6 z^4 a^{-6} -8 z^2 a^{-2} +25 z^2 a^{-4} -13 z^2 a^{-6} +z^2 a^{-8} -4 a^{-2} +13 a^{-4} -10 a^{-6} +2 a^{-8} }
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +6 z^8 a^{-4} +4 z^8 a^{-6} +z^7 a^{-1} +z^7 a^{-3} +5 z^7 a^{-5} +5 z^7 a^{-7} -9 z^6 a^{-2} -24 z^6 a^{-4} -13 z^6 a^{-6} +2 z^6 a^{-8} -5 z^5 a^{-1} -18 z^5 a^{-3} -31 z^5 a^{-5} -18 z^5 a^{-7} +13 z^4 a^{-2} +31 z^4 a^{-4} +16 z^4 a^{-6} -2 z^4 a^{-8} +8 z^3 a^{-1} +28 z^3 a^{-3} +42 z^3 a^{-5} +26 z^3 a^{-7} +4 z^3 a^{-9} -9 z^2 a^{-2} -26 z^2 a^{-4} -17 z^2 a^{-6} +z^2 a^{-8} +z^2 a^{-10} -4 z a^{-1} -13 z a^{-3} -21 z a^{-5} -14 z a^{-7} -2 z a^{-9} +4 a^{-2} +13 a^{-4} +10 a^{-6} +2 a^{-8} }
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^2-2 q^{-2} + q^{-6} +2 q^{-8} +6 q^{-10} +2 q^{-12} +4 q^{-14} -2 q^{-16} -3 q^{-18} -3 q^{-20} -3 q^{-22} + q^{-24} + q^{-28} }
The G2 invariant	Data:K11n78/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n78 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["K11n78"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-3t^{3}+6t^{2}-8t+9-8t^{-1}+6t^{-2}-3t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+5z^{6}+8z^{4}+5z^{2}+1$

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]=

{ 45, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^8-4 q^7+5 q^6-7 q^5+8 q^4-6 q^3+7 q^2-4 q+2- q^{-1} }

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +19 z^4 a^{-4} -6 z^4 a^{-6} -8 z^2 a^{-2} +25 z^2 a^{-4} -13 z^2 a^{-6} +z^2 a^{-8} -4 a^{-2} +13 a^{-4} -10 a^{-6} +2 a^{-8} }

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +6 z^8 a^{-4} +4 z^8 a^{-6} +z^7 a^{-1} +z^7 a^{-3} +5 z^7 a^{-5} +5 z^7 a^{-7} -9 z^6 a^{-2} -24 z^6 a^{-4} -13 z^6 a^{-6} +2 z^6 a^{-8} -5 z^5 a^{-1} -18 z^5 a^{-3} -31 z^5 a^{-5} -18 z^5 a^{-7} +13 z^4 a^{-2} +31 z^4 a^{-4} +16 z^4 a^{-6} -2 z^4 a^{-8} +8 z^3 a^{-1} +28 z^3 a^{-3} +42 z^3 a^{-5} +26 z^3 a^{-7} +4 z^3 a^{-9} -9 z^2 a^{-2} -26 z^2 a^{-4} -17 z^2 a^{-6} +z^2 a^{-8} +z^2 a^{-10} -4 z a^{-1} -13 z a^{-3} -21 z a^{-5} -14 z a^{-7} -2 z a^{-9} +4 a^{-2} +13 a^{-4} +10 a^{-6} +2 a^{-8} }

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_62, K11n76,}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {K11n76,}

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["K11n78"];

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]=

{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^4-3 t^3+6 t^2-8 t+9-8 t^{-1} +6 t^{-2} -3 t^{-3} + t^{-4} } , $q^{8}-4q^{7}+5q^{6}-7q^{5}+8q^{4}-6q^{3}+7q^{2}-4q+2-q^{-1}$ }

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]=

{10_62, K11n76,}

In[6]:=

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

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

Out[6]=

{K11n76,}

Vassiliev invariants

V2 and V3:

(5, 7)

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 20} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 56} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 200} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{886}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{98}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1120} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{4496}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{704}{3}} $152$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{4000}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1568} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{17720}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1960}{3}} ${\frac {50767}{6}}$ ${\frac {2090}{3}}$ ${\frac {21446}{9}}$ ${\frac {853}{18}}$ ${\frac {1423}{6}}$

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 K11n78. 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 χ 17                   1 1 15                 3   -3 13               2 1   1 11             5 3     -2 9           3 2       1 7         3 5         2 5       4 3           1 3     1 4             3 1   1 3               -2 -1   1                 1 -3 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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.

K11n77 K11n79