K11a362

K11a361

K11a363

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a362 at Knotilus!
	Also known as the pretzel knot P(5,3,3).

Knot presentations

Planar diagram presentation	X8291 X14,4,15,3 X18,6,19,5 X16,8,17,7 X22,10,1,9 X20,12,21,11 X4,14,5,13 X2,16,3,15 X6,18,7,17 X12,20,13,19 X10,22,11,21
Gauss code	1, -8, 2, -7, 3, -9, 4, -1, 5, -11, 6, -10, 7, -2, 8, -4, 9, -3, 10, -6, 11, -5
Dowker-Thistlethwaite code	8 14 18 16 22 20 4 2 6 12 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2,3\}$ 3-genus 1 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a362/ThurstonBennequinNumber Hyperbolic Volume 8.76519 A-Polynomial See Data:K11a362/A-polynomial

[edit Notes for K11a362'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 K11a362's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$10t-19+10t^{-1}$
Conway polynomial	$10z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 39, 2 }
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}+q^{11}-3 q^{10}+4 q^9-4 q^8+6 q^7-5 q^6+5 q^5-4 q^4+3 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^2 a^{-2} +2 z^2 a^{-4} +3 z^2 a^{-6} +3 z^2 a^{-8} +z^2 a^{-10} + a^{-6} +2 a^{-8} - a^{-10} - a^{-12} }
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^{-10} +z^{10} a^{-12} +3 z^9 a^{-9} +4 z^9 a^{-11} +z^9 a^{-13} +5 z^8 a^{-8} -z^8 a^{-10} -6 z^8 a^{-12} +5 z^7 a^{-7} -13 z^7 a^{-9} -26 z^7 a^{-11} -8 z^7 a^{-13} +5 z^6 a^{-6} -22 z^6 a^{-8} -17 z^6 a^{-10} +10 z^6 a^{-12} +4 z^5 a^{-5} -16 z^5 a^{-7} +13 z^5 a^{-9} +56 z^5 a^{-11} +23 z^5 a^{-13} +3 z^4 a^{-4} -13 z^4 a^{-6} +27 z^4 a^{-8} +41 z^4 a^{-10} -2 z^4 a^{-12} +2 z^3 a^{-3} -6 z^3 a^{-5} +9 z^3 a^{-7} -2 z^3 a^{-9} -47 z^3 a^{-11} -28 z^3 a^{-13} +z^2 a^{-2} -2 z^2 a^{-4} +6 z^2 a^{-6} -12 z^2 a^{-8} -24 z^2 a^{-10} -3 z^2 a^{-12} +2 z a^{-9} +14 z a^{-11} +12 z a^{-13} - a^{-6} +2 a^{-8} + a^{-10} - a^{-12} }
The A2 invariant	Data:K11a362/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a362/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a362 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["K11a362"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$10t-19+10t^{-1}$

In[5]:=

Conway[K][z]

Out[5]=

$10z^{2}+1$

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]=

{ 39, 2 }

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

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^{-10} +z^{10} a^{-12} +3 z^9 a^{-9} +4 z^9 a^{-11} +z^9 a^{-13} +5 z^8 a^{-8} -z^8 a^{-10} -6 z^8 a^{-12} +5 z^7 a^{-7} -13 z^7 a^{-9} -26 z^7 a^{-11} -8 z^7 a^{-13} +5 z^6 a^{-6} -22 z^6 a^{-8} -17 z^6 a^{-10} +10 z^6 a^{-12} +4 z^5 a^{-5} -16 z^5 a^{-7} +13 z^5 a^{-9} +56 z^5 a^{-11} +23 z^5 a^{-13} +3 z^4 a^{-4} -13 z^4 a^{-6} +27 z^4 a^{-8} +41 z^4 a^{-10} -2 z^4 a^{-12} +2 z^3 a^{-3} -6 z^3 a^{-5} +9 z^3 a^{-7} -2 z^3 a^{-9} -47 z^3 a^{-11} -28 z^3 a^{-13} +z^2 a^{-2} -2 z^2 a^{-4} +6 z^2 a^{-6} -12 z^2 a^{-8} -24 z^2 a^{-10} -3 z^2 a^{-12} +2 z a^{-9} +14 z a^{-11} +12 z a^{-13} - a^{-6} +2 a^{-8} + a^{-10} - a^{-12} }

"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["K11a362"];

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 10 t-19+10 t^{-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 -q^{12}+q^{11}-3 q^{10}+4 q^9-4 q^8+6 q^7-5 q^6+5 q^5-4 q^4+3 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:

(10, 31)

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 40} 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 248} $800$ 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{6236}{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{1180}{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 9920} ${\frac {56240}{3}}$ ${\frac {9920}{3}}$ $3224$ ${\frac {32000}{3}}$ $30752$ ${\frac {249440}{3}}$ ${\frac {47200}{3}}$ ${\frac {509623}{3}}$ $-{\frac {15892}{3}}$ ${\frac {732964}{9}}$ ${\frac {13405}{9}}$ ${\frac {35671}{3}}$

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 K11a362. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 χ 25                       1 -1 23                         0 21                   3 1   -2 19                 1       1 17               3 3       0 15             3 1         2 13           2 3           1 11         3 3             0 9       1 2               1 7     2 3                 -1 5     1                   1 3 1 2                     -1 1 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ 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}^{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 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}^{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}} 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^{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}^{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 r=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 {\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=11} 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.

K11a361 K11a363