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L11n298 L11n300

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X13,19,14,18 X17,11,18,22 X7,17,8,16 X21,8,22,9 X9,14,10,15 X15,20,16,21 X19,5,20,10 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-7 + 3q-6 - 3q-5 + 4q-4 - 2q-3 + 3q-2 + q - 2q2 + q3

A2 (sl(3)) Invariant:

- q-22 + 2q-18 + 3q-16 + 5q-14 + 4q-12 + 6q-10 + 4q-8 + 2q-6 + 2q-4 - q-2 + 1 - q8 + q10

HOMFLY-PT Polynomial:

a-2z2 + 1 + z2 + a2z-2 - a2 - 6a2z2 - 5a2z4 - a2z6 - 2a4z-2 + a4 + 6a4z2 + 2a4z4 + a6z-2 - a6 - a6z2

Kauffman Polynomial:

2a-2z2 - 4a-2z4 + a-2z6 - 2a-1z + 10a-1z3 - 10a-1z5 + 2a-1z7 + 2 - 7z2 + 8z4 - 6z6 + z8 - 7az + 21az3 - 15az5 + 2az7 + a2z-2 + 4a2 - 30a2z2 + 48a2z4 - 27a2z6 + 4a2z8 - 2a3z-1 - 7a3z + 18a3z3 - 9a3z7 + 2a3z9 + 2a4z-2 + 3a4 - 29a4z2 + 55a4z4 - 35a4z6 + 6a4z8 - 2a5z-1 - 3a5z + 10a5z3 + a5z5 - 8a5z7 + 2a5z9 + a6z-2 - 8a6z2 + 19a6z4 - 15a6z6 + 3a6z8 - a7z + 3a7z3 - 4a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     1   j = 3                 1 1 1   j = 1               2 2 1     j = -1             2 3 1       j = -3           3 3 3         j = -5         3 5 1           j = -7       2 2 2             j = -9     2 4 1               j = -11   1 1                   j = -13   2                     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, 299]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 299]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[13, 19, 14, 18], X[17, 11, 18, 22], > X[7, 17, 8, 16], X[21, 8, 22, 9], X[9, 14, 10, 15], X[15, 20, 16, 21], > X[19, 5, 20, 10], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -5, 6, -7, 9}, > {11, -2, -3, 7, -8, 5, -4, 3, -9, 8, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-7 3 3 4 2 3 2 3 -q + -- - -- + -- - -- + -- + q - 2 q + q 6 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 3 5 4 6 4 2 2 -2 8 10 1 - q + --- + --- + --- + --- + --- + -- + -- + -- - q - q + q 18 16 14 12 10 8 6 4 q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 299]][a, z]
Out[8]=	2 4 6 2 2 4 6 a 2 a a 2 z 2 2 4 2 6 2 1 - a + a - a + -- - ---- + -- + z + -- - 6 a z + 6 a z - a z - 2 2 2 2 z z z a 2 4 4 4 2 6 > 5 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 299]][a, z]
Out[9]=	2 4 6 3 5 2 4 a 2 a a 2 a 2 a 2 z 3 2 + 4 a + 3 a + -- + ---- + -- - ---- - ---- - --- - 7 a z - 7 a z - 2 2 2 z z a z z z 2 3 5 7 2 2 z 2 2 4 2 6 2 10 z > 3 a z - a z - 7 z + ---- - 30 a z - 29 a z - 8 a z + ----- + 2 a a 4 3 3 3 5 3 7 3 4 4 z 2 4 > 21 a z + 18 a z + 10 a z + 3 a z + 8 z - ---- + 48 a z + 2 a 5 6 4 4 6 4 10 z 5 5 5 7 5 6 z > 55 a z + 19 a z - ----- - 15 a z + a z - 4 a z - 6 z + -- - a 2 a 7 2 6 4 6 6 6 2 z 7 3 7 5 7 > 27 a z - 35 a z - 15 a z + ---- + 2 a z - 9 a z - 8 a z + a 7 7 8 2 8 4 8 6 8 3 9 5 9 > a z + z + 4 a z + 6 a z + 3 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 2 1 1 2 4 2 -- + - + 2 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 q q t q t q t q t q t q t q t 1 2 3 2 5 3 1 3 2 t > ----- + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + - + 9 3 7 3 5 3 7 2 5 2 3 2 5 3 q t q q t q t q t q t q t q t q t q t 3 2 3 2 3 3 5 3 7 4 > 2 q t + q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n299

L11n298 L11n300