K11a308

K11a307

K11a309

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a308 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X12,4,13,3 X14,6,15,5 X18,8,19,7 X20,10,21,9 X2,12,3,11 X4,14,5,13 X22,15,1,16 X10,18,11,17 X8,20,9,19 X16,21,17,22
Gauss code	1, -6, 2, -7, 3, -1, 4, -10, 5, -9, 6, -2, 7, -3, 8, -11, 9, -4, 10, -5, 11, -8
Dowker-Thistlethwaite code	6 12 14 18 20 2 4 22 10 8 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a308/ThurstonBennequinNumber Hyperbolic Volume 10.8655 A-Polynomial See Data:K11a308/A-polynomial

[edit Notes for K11a308's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus 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 4} Rasmussen s-Invariant -6

[edit Notes for K11a308's four dimensional invariants]

Polynomial invariants

Alexander 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 -t^4+5 t^3-9 t^2+13 t-15+13 t^{-1} -9 t^{-2} +5 t^{-3} - t^{-4} }
Conway 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 -z^8-3 z^6+z^4+6 z^2+1}
2nd Alexander ideal (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 \{1\}}
Determinant and Signature	{ 71, 6 }
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^{12}+3 q^{11}-6 q^{10}+8 q^9-10 q^8+11 q^7-10 q^6+9 q^5-6 q^4+4 q^3-2 q^2+q}
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^{-6} +z^6 a^{-4} -6 z^6 a^{-6} +2 z^6 a^{-8} +5 z^4 a^{-4} -12 z^4 a^{-6} +9 z^4 a^{-8} -z^4 a^{-10} +7 z^2 a^{-4} -9 z^2 a^{-6} +11 z^2 a^{-8} -3 z^2 a^{-10} +2 a^{-4} -2 a^{-6} +3 a^{-8} -2 a^{-10} }
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^{10} a^{-6} +z^{10} a^{-8} +2 z^9 a^{-5} +5 z^9 a^{-7} +3 z^9 a^{-9} +z^8 a^{-4} -z^8 a^{-6} +4 z^8 a^{-8} +6 z^8 a^{-10} -11 z^7 a^{-5} -20 z^7 a^{-7} -z^7 a^{-9} +8 z^7 a^{-11} -6 z^6 a^{-4} -12 z^6 a^{-6} -25 z^6 a^{-8} -11 z^6 a^{-10} +8 z^6 a^{-12} +19 z^5 a^{-5} +21 z^5 a^{-7} -18 z^5 a^{-9} -14 z^5 a^{-11} +6 z^5 a^{-13} +12 z^4 a^{-4} +26 z^4 a^{-6} +30 z^4 a^{-8} +2 z^4 a^{-10} -11 z^4 a^{-12} +3 z^4 a^{-14} -10 z^3 a^{-5} -4 z^3 a^{-7} +17 z^3 a^{-9} +5 z^3 a^{-11} -5 z^3 a^{-13} +z^3 a^{-15} -9 z^2 a^{-4} -14 z^2 a^{-6} -13 z^2 a^{-8} -4 z^2 a^{-10} +4 z^2 a^{-12} -4 z a^{-9} -2 z a^{-11} +2 z a^{-13} +2 a^{-4} +2 a^{-6} +3 a^{-8} +2 a^{-10} }
The A2 invariant	$q^{-4}+q^{-8}-q^{-12}+2q^{-14}-q^{-16}+3q^{-18}+q^{-24}-2q^{-26}+q^{-28}-q^{-30}-q^{-36}$
The G2 invariant	Data:K11a308/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a308 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["K11a308"];

In[4]:=

Alexander[K][t]

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

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+5 t^3-9 t^2+13 t-15+13 t^{-1} -9 t^{-2} +5 t^{-3} - t^{-4} }

In[5]:=

Conway[K][z]

Out[5]=

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^8-3 z^6+z^4+6 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]=

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 \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 71, 6 }

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

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^{-6} +z^6 a^{-4} -6 z^6 a^{-6} +2 z^6 a^{-8} +5 z^4 a^{-4} -12 z^4 a^{-6} +9 z^4 a^{-8} -z^4 a^{-10} +7 z^2 a^{-4} -9 z^2 a^{-6} +11 z^2 a^{-8} -3 z^2 a^{-10} +2 a^{-4} -2 a^{-6} +3 a^{-8} -2 a^{-10} }

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^{10} a^{-6} +z^{10} a^{-8} +2 z^9 a^{-5} +5 z^9 a^{-7} +3 z^9 a^{-9} +z^8 a^{-4} -z^8 a^{-6} +4 z^8 a^{-8} +6 z^8 a^{-10} -11 z^7 a^{-5} -20 z^7 a^{-7} -z^7 a^{-9} +8 z^7 a^{-11} -6 z^6 a^{-4} -12 z^6 a^{-6} -25 z^6 a^{-8} -11 z^6 a^{-10} +8 z^6 a^{-12} +19 z^5 a^{-5} +21 z^5 a^{-7} -18 z^5 a^{-9} -14 z^5 a^{-11} +6 z^5 a^{-13} +12 z^4 a^{-4} +26 z^4 a^{-6} +30 z^4 a^{-8} +2 z^4 a^{-10} -11 z^4 a^{-12} +3 z^4 a^{-14} -10 z^3 a^{-5} -4 z^3 a^{-7} +17 z^3 a^{-9} +5 z^3 a^{-11} -5 z^3 a^{-13} +z^3 a^{-15} -9 z^2 a^{-4} -14 z^2 a^{-6} -13 z^2 a^{-8} -4 z^2 a^{-10} +4 z^2 a^{-12} -4 z a^{-9} -2 z a^{-11} +2 z a^{-13} +2 a^{-4} +2 a^{-6} +3 a^{-8} +2 a^{-10} }

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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}} ): {}

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["K11a308"];

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+5 t^3-9 t^2+13 t-15+13 t^{-1} -9 t^{-2} +5 t^{-3} - t^{-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 -q^{12}+3 q^{11}-6 q^{10}+8 q^9-10 q^8+11 q^7-10 q^6+9 q^5-6 q^4+4 q^3-2 q^2+q} }

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]=

{}

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:

(6, 16)

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 $24$ $128$ $288$ $828$ Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 132} $3072$ ${\frac {17792}{3}}$ ${\frac {2912}{3}}$ $928$ $2304$ $8192$ $19872$ $3168$ ${\frac {215311}{5}}$ $-{\frac {2204}{5}}$ ${\frac {278204}{15}}$ Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {1681}{3}}} ${\frac {12431}{5}}$

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 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 r} ). The squares with yellow highlighting are those on the "critical diagonals", where 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 j-2r=s+1} or 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 j-2r=s-1} , where 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 s=} 6 is the signature of K11a308. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 25                       1 -1 23                     2   2 21                   4 1   -3 19                 4 2     2 17               6 4       -2 15             5 4         1 13           5 6           1 11         4 5             -1 9       2 5               3 7     2 4                 -2 5   1 3                   2 3   1                     -1 1 1                       1

Integral Khovanov Homology (db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} 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 i=5} 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 i=7} 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 r=-2} ${\mathbb {Z} }$ 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 r=-1} 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}\oplus{\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}} 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 r=0} 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}^{3}\oplus{\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}^{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 r=1} 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}^{4}\oplus{\mathbb Z}_2^{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}^{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 r=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}^{5}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}} 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}^{5}} 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 r=4} ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ 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}^{5}} 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 r=5} 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}^{4}\oplus{\mathbb Z}_2^{6}} 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}^{6}} 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 r=6} 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}^{4}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=7} 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}^{2}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}\oplus{\mathbb Z}_2^{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}^{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 r=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 {\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.

K11a307 K11a309