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L11n303 L11n305

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X20,13,21,14 X22,19,11,20 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 X14,21,15,22 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-7 + 2q-6 - 4q-5 + 6q-4 - 6q-3 + 8q-2 - 6q-1 + 6 - 3q + 2q2

A2 (sl(3)) Invariant:

- q-22 - 2q-20 - q-18 - 2q-16 + q-14 + q-12 + 2q-10 + 5q-8 + 3q-6 + 6q-4 + 4q-2 + 4 + 4q2 + q4 + 2q6

HOMFLY-PT Polynomial:

2z-2 + 7 + 7z2 + 2z4 - 5a2z-2 - 17a2 - 21a2z2 - 11a2z4 - 2a2z6 + 4a4z-2 + 13a4 + 12a4z2 + 3a4z4 - a6z-2 - 3a6 - a6z2

Kauffman Polynomial:

- 2a-2 + 3a-2z2 + a-1z + a-1z3 + a-1z5 - 2z-2 + 6 - 4z2 + 2z6 + 5az-1 - 16az + 18az3 - 12az5 + 4az7 - 5a2z-2 + 20a2 - 32a2z2 + 26a2z4 - 15a2z6 + 4a2z8 + 9a3z-1 - 33a3z + 44a3z3 - 28a3z5 + 4a3z7 + a3z9 - 4a4z-2 + 17a4 - 32a4z2 + 38a4z4 - 26a4z6 + 6a4z8 + 5a5z-1 - 21a5z + 35a5z3 - 20a5z5 + a5z7 + a5z9 - a6z-2 + 4a6 - 7a6z2 + 12a6z4 - 9a6z6 + 2a6z8 + a7z-1 - 5a7z + 8a7z3 - 5a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 5                   2 j = 3                 3 2 j = 1               3     j = -1             4 4     j = -3           4 2       j = -5         2 4         j = -7       4 4           j = -9     1 3             j = -11   1 3               j = -13   1                 j = -15 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, 304]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 304]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 13, 21, 14], X[22, 19, 11, 20], > X[15, 5, 16, 10], X[17, 9, 18, 8], X[7, 17, 8, 16], X[9, 19, 10, 18], > X[14, 21, 15, 22], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -7, 6, -8, 5}, > {11, -2, 3, -9, -5, 7, -6, 8, 4, -3, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-7 2 4 6 6 8 6 2 6 - q + -- - -- + -- - -- + -- - - - 3 q + 2 q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 -18 2 -14 -12 2 5 3 6 4 2 4 - q - --- - q - --- + q + q + --- + -- + -- + -- + -- + 4 q + 20 16 10 8 6 4 2 q q q q q q q 4 6 > q + 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 304]][a, z]
Out[8]=	2 4 6 2 4 6 2 5 a 4 a a 2 2 2 4 2 7 - 17 a + 13 a - 3 a + -- - ---- + ---- - -- + 7 z - 21 a z + 12 a z - 2 2 2 2 z z z z 6 2 4 2 4 4 4 2 6 > a z + 2 z - 11 a z + 3 a z - 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 304]][a, z]
Out[9]=	2 4 6 3 5 2 2 4 6 2 5 a 4 a a 5 a 9 a 5 a 6 - -- + 20 a + 17 a + 4 a - -- - ---- - ---- - -- + --- + ---- + ---- + 2 2 2 2 2 z z z a z z z z 7 2 a z 3 5 7 2 3 z 2 2 > -- + - - 16 a z - 33 a z - 21 a z - 5 a z - 4 z + ---- - 32 a z - z a 2 a 3 4 2 6 2 z 3 3 3 5 3 7 3 > 32 a z - 7 a z + -- + 18 a z + 44 a z + 35 a z + 8 a z + a 5 2 4 4 4 6 4 z 5 3 5 5 5 > 26 a z + 38 a z + 12 a z + -- - 12 a z - 28 a z - 20 a z - a 7 5 6 2 6 4 6 6 6 7 3 7 5 7 > 5 a z + 2 z - 15 a z - 26 a z - 9 a z + 4 a z + 4 a z + a z + 7 7 2 8 4 8 6 8 3 9 5 9 > a z + 4 a z + 6 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 1 3 1 3 4 4 - + 3 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 2 4 4 2 4 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 3 q t + 2 q t + 2 q t 5 3 5 2 3 2 3 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n304

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