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L11n257 L11n259

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,18,12,19 X19,22,20,9 X15,20,16,21 X21,16,22,17 X17,12,18,13 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

- 3q-6 + 6q-5 - 10q-4 + 13q-3 - 12q-2 + 13q-1 - 9 + 7q - 2q2 + q3

A2 (sl(3)) Invariant:

- 2q-20 - 4q-18 - 4q-14 - q-12 + 2q-10 + 7q-6 + 3q-4 + 8q-2 + 6 + 3q2 + 6q4 + q6 + q8 + q10

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 + a-2z2 - z-2 - 3 - 4z2 - 2z4 - 2a2z-2 - 3a2 + 2a2z4 + a2z6 + 3a4z-2 + 6a4 + 2a4z2 - a4z4 - a6z-2 - 2a6

Kauffman Polynomial:

a-2z-2 - 4a-2 + 6a-2z2 - 4a-2z4 + a-2z6 - 2a-1z-1 + 4a-1z - 4a-1z5 + 2a-1z7 + z-2 - 3 + 4z2 - z4 - 5z6 + 3z8 + 2az-1 - 10az + 25az3 - 30az5 + 9az7 + az9 - 2a2z-2 + 4a2 - 2a2z2 + a2z4 - 13a2z6 + 8a2z8 + 10a3z-1 - 34a3z + 51a3z3 - 43a3z5 + 14a3z7 + a3z9 - 3a4z-2 + 5a4 + a4z2 - 2a4z4 - 4a4z6 + 5a4z8 + 8a5z-1 - 27a5z + 32a5z3 - 17a5z5 + 7a5z7 - a6z-2 + a6 + a6z2 + 3a6z6 + 2a7z-1 - 7a7z + 6a7z3

Khovanov Homology:

trqj 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                 2 1 j = 3               5     j = 1             4 2     j = -1           9 5       j = -3         6 7         j = -5       7 6           j = -7     3 6             j = -9   3 7               j = -11   3                 j = -13 3

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, 258]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 258]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 15, 8, 14], X[13, 5, 14, 8], > X[11, 18, 12, 19], X[19, 22, 20, 9], X[15, 20, 16, 21], X[21, 16, 22, 17], > X[17, 12, 18, 13], X[2, 5, 3, 6], X[9, 1, 10, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 4}, > {-11, 2, -5, 9, -4, 3, -7, 8, -9, 5, -6, 7, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	3 6 10 13 12 13 2 3 -9 - -- + -- - -- + -- - -- + -- + 7 q - 2 q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 4 -12 2 7 3 8 2 4 6 8 10 6 - --- - --- - --- - q + --- + -- + -- + -- + 3 q + 6 q + q + q + q 20 18 14 10 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 258]][a, z]
Out[8]=	2 4 6 2 2 2 4 6 -2 1 2 a 3 a a 2 z -3 + -- - 3 a + 6 a - 2 a - z + ----- - ---- + ---- - -- - 4 z + -- + 2 2 2 2 2 2 2 a a z z z z a 4 2 4 2 4 4 4 2 6 > 2 a z - 2 z + 2 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 258]][a, z]
Out[9]=	2 4 6 4 2 4 6 -2 1 2 a 3 a a 2 2 a -3 - -- + 4 a + 5 a + a + z + ----- - ---- - ---- - -- - --- + --- + 2 2 2 2 2 2 a z z a a z z z z 3 5 7 10 a 8 a 2 a 4 z 3 5 7 2 > ----- + ---- + ---- + --- - 10 a z - 34 a z - 27 a z - 7 a z + 4 z + z z z a 2 6 z 2 2 4 2 6 2 3 3 3 5 3 7 3 > ---- - 2 a z + a z + a z + 25 a z + 51 a z + 32 a z + 6 a z - 2 a 4 5 4 4 z 2 4 4 4 4 z 5 3 5 5 5 6 > z - ---- + a z - 2 a z - ---- - 30 a z - 43 a z - 17 a z - 5 z + 2 a a 6 7 z 2 6 4 6 6 6 2 z 7 3 7 5 7 > -- - 13 a z - 4 a z + 3 a z + ---- + 9 a z + 14 a z + 7 a z + 2 a a 8 2 8 4 8 9 3 9 > 3 z + 8 a z + 5 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 9 3 3 3 7 3 6 7 6 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 6 5 t 2 3 2 5 3 5 4 7 4 > ---- + --- + 4 q t + 2 q t + 5 q t + 2 q t + q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n258

L11n257 L11n259