© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a137 K11a139

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X16,5,17,6 X14,8,15,7 X18,10,19,9 X2,11,3,12 X20,14,21,13 X22,15,1,16 X12,18,13,17 X8,20,9,19 X6,21,7,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 14 18 2 20 22 12 8 6

Alexander Polynomial:

- 2t-3 + 14t-2 - 38t-1 + 53 - 38t + 14t2 - 2t3

Conway Polynomial:

1 + 2z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a285, ...}

Determinant and Signature:

{161, 0}

Jones Polynomial:

- q-5 + 4q-4 - 9q-3 + 16q-2 - 22q-1 + 26 - 26q + 23q2 - 17q3 + 11q4 - 5q5 + q6

Other knots (up to mirrors) with the same Jones Polynomial:

{K11a285, ...}

A2 (sl(3)) Invariant:

- q-16 + q-14 + 2q-12 - 4q-10 + 3q-8 + q-6 - 4q-4 + 5q-2 - 3 + 3q2 - 2q6 + 5q8 - 5q10 + 2q12 + 2q14 - 3q16 + q18

HOMFLY-PT Polynomial:

a-4z4 - a-2z2 - a-2z4 - a-2z6 + 1 + z2 - z6 + a2z2 + 2a2z4 - a4z2

Kauffman Polynomial:

- a-6z4 + a-6z6 + 3a-5z3 - 9a-5z5 + 5a-5z7 - a-4z2 + 12a-4z4 - 22a-4z6 + 10a-4z8 - a-3z + 8a-3z3 - 6a-3z5 - 12a-3z7 + 9a-3z9 - 7a-2z2 + 35a-2z4 - 49a-2z6 + 16a-2z8 + 3a-2z10 - 3a-1z + 10a-1z3 + 2a-1z5 - 27a-1z7 + 17a-1z9 + 1 - 9z2 + 32z4 - 42z6 + 16z8 + 3z10 - 3az + 12az3 - 13az5 - 2az7 + 8az9 - a2z2 + 5a2z4 - 12a2z6 + 10a2z8 - a3z + 6a3z3 - 11a3z5 + 8a3z7 + 2a4z2 - 5a4z4 + 4a4z6 - a5z3 + a5z5

V2 and V3, the type 2 and 3 Vassiliev invariants:

{0, 0}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 11138. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     4   j = 9                   7 1   j = 7                 10 4     j = 5               13 7       j = 3             13 10         j = 1           13 13           j = -1         10 14             j = -3       6 12               j = -5     3 10                 j = -7   1 6                   j = -9   3                     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]:=	Show[DrawMorseLink[Knot[11, Alternating, 138]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 138]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[16, 5, 17, 6], X[14, 8, 15, 7], > X[18, 10, 19, 9], X[2, 11, 3, 12], X[20, 14, 21, 13], X[22, 15, 1, 16], > X[12, 18, 13, 17], X[8, 20, 9, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 138]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -10, 5, -2, 6, -9, 7, -4, 8, -3, 9, -5, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 138]]
Out[5]=	DTCode[4, 10, 16, 14, 18, 2, 20, 22, 12, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 138]][t]
Out[6]=	2 14 38 2 3 53 - -- + -- - -- - 38 t + 14 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 138]][z]
Out[7]=	4 6 1 + 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 138], Knot[11, Alternating, 285]}
In[9]:=	{KnotDet[Knot[11, Alternating, 138]], KnotSignature[Knot[11, Alternating, 138]]}
Out[9]=	{161, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 138]][q]
Out[10]=	-5 4 9 16 22 2 3 4 5 6 26 - q + -- - -- + -- - -- - 26 q + 23 q - 17 q + 11 q - 5 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 138], Knot[11, Alternating, 285]}
In[12]:=	A2Invariant[Knot[11, Alternating, 138]][q]
Out[12]=	-16 -14 2 4 3 -6 4 5 2 6 8 -3 - q + q + --- - --- + -- + q - -- + -- + 3 q - 2 q + 5 q - 12 10 8 4 2 q q q q q 10 12 14 16 18 > 5 q + 2 q + 2 q - 3 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 138]][a, z]
Out[13]=	2 4 4 6 2 z 2 2 4 2 z z 2 4 6 z 1 + z - -- + a z - a z + -- - -- + 2 a z - z - -- 2 4 2 2 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 138]][a, z]
Out[14]=	2 2 3 z 3 z 3 2 z 7 z 2 2 4 2 3 z 1 - -- - --- - 3 a z - a z - 9 z - -- - ---- - a z + 2 a z + ---- + 3 a 4 2 5 a a a a 3 3 4 4 4 8 z 10 z 3 3 3 5 3 4 z 12 z 35 z > ---- + ----- + 12 a z + 6 a z - a z + 32 z - -- + ----- + ----- + 3 a 6 4 2 a a a a 5 5 5 2 4 4 4 9 z 6 z 2 z 5 3 5 5 5 > 5 a z - 5 a z - ---- - ---- + ---- - 13 a z - 11 a z + a z - 5 3 a a a 6 6 6 7 7 7 6 z 22 z 49 z 2 6 4 6 5 z 12 z 27 z > 42 z + -- - ----- - ----- - 12 a z + 4 a z + ---- - ----- - ----- - 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 10 z 16 z 2 8 9 z 17 z > 2 a z + 8 a z + 16 z + ----- + ----- + 10 a z + ---- + ----- + 4 2 3 a a a a 10 9 10 3 z > 8 a z + 3 z + ----- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 138]], Vassiliev[3][Knot[11, Alternating, 138]]}
Out[15]=	{0, 0}
In[16]:=	Kh[Knot[11, Alternating, 138]][q, t]
Out[16]=	14 1 3 1 6 3 10 6 12 -- + 13 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 10 3 3 2 5 2 5 3 7 3 > --- + 13 q t + 13 q t + 10 q t + 13 q t + 7 q t + 10 q t + q t 7 4 9 4 9 5 11 5 13 6 > 4 q t + 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a138

K11a137 K11a139