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L11n330 L11n332

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DrawMorseLink

PD Presentation:

X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X13,18,14,19 X9,13,10,22 X11,21,12,20 X19,5,20,12 X21,11,22,10 X4,17,1,18

Gauss Code:

{{1, -4, 3, -11}, {-2, -1, 5, -3, -7, 10, -8, 9}, {-6, 2, 4, -5, 11, 6, -9, 8, -10, 7}}

Jones Polynomial:

- q-5 + 2q-4 - 3q-3 + 5q-2 - 5q-1 + 6 - 4q + 4q2 - q3 + q4

A2 (sl(3)) Invariant:

- q-16 - q-10 + 2q-8 + q-6 + q-4 + 2q-2 + 1 + 4q2 + 3q4 + 5q6 + 5q8 + 2q10 + 2q12 + q14

HOMFLY-PT Polynomial:

a-4z-2 + a-4 - 2a-2z-2 - 3a-2 - 2a-2z2 + z-2 + 1 + z2 + z4 + 2a2 + 2a2z2 + a2z4 - a4 - a4z2

Kauffman Polynomial:

- a-4z-2 + 4a-4 - 4a-4z2 + a-4z4 + 2a-3z-1 - 3a-3z - a-3z3 + a-3z5 - 2a-2z-2 + 9a-2 - 20a-2z2 + 17a-2z4 - 6a-2z6 + a-2z8 + 2a-1z-1 - 6a-1z + 9a-1z5 - 5a-1z7 + a-1z9 - z-2 + 7 - 21z2 + 30z4 - 15z6 + 3z8 - 3az + 5az3 + 2az5 - 3az7 + az9 - 2a2z2 + 8a2z4 - 7a2z6 + 2a2z8 + a3z + a3z3 - 5a3z5 + 2a3z7 - a4 + 3a4z2 - 6a4z4 + 2a4z6 + a5z - 3a5z3 + a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                   1 j = 7                     j = 5               4 1   j = 3             2 2     j = 1           4 2       j = -1         3 4         j = -3       2 2           j = -5     1 3             j = -7   1 2               j = -9   1                 j = -11 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 331]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 331]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[13, 18, 14, 19], X[9, 13, 10, 22], X[11, 21, 12, 20], > X[19, 5, 20, 12], X[21, 11, 22, 10], X[4, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11}, {-2, -1, 5, -3, -7, 10, -8, 9}, > {-6, 2, 4, -5, 11, 6, -9, 8, -10, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-5 2 3 5 5 2 3 4 6 - q + -- - -- + -- - - - 4 q + 4 q - q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -10 2 -6 -4 2 2 4 6 8 10 1 - q - q + -- + q + q + -- + 4 q + 3 q + 5 q + 5 q + 2 q + 8 2 q q 12 14 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 331]][a, z]
Out[8]=	2 -4 3 2 4 -2 1 2 2 2 z 2 2 4 2 1 + a - -- + 2 a - a + z + ----- - ----- + z - ---- + 2 a z - a z + 2 4 2 2 2 2 a a z a z a 4 2 4 > z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 331]][a, z]
Out[9]=	4 9 4 -2 1 2 2 2 3 z 6 z 7 + -- + -- - a - z - ----- - ----- + ---- + --- - --- - --- - 3 a z + 4 2 4 2 2 2 3 a z 3 a a a a z a z a z a 2 2 3 3 5 2 4 z 20 z 2 2 4 2 z 3 > a z + a z - 21 z - ---- - ----- - 2 a z + 3 a z - -- + 5 a z + 4 2 3 a a a 4 4 5 5 3 3 5 3 4 z 17 z 2 4 4 4 z 9 z > a z - 3 a z + 30 z + -- + ----- + 8 a z - 6 a z + -- + ---- + 4 2 3 a a a a 6 7 5 3 5 5 5 6 6 z 2 6 4 6 5 z > 2 a z - 5 a z + a z - 15 z - ---- - 7 a z + 2 a z - ---- - 2 a a 8 9 7 3 7 8 z 2 8 z 9 > 3 a z + 2 a z + 3 z + -- + 2 a z + -- + a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 1 2 1 3 2 2 3 - + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 9 4 > 2 q t + 2 q t + 2 q t + 4 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n331

L11n330 L11n332