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

K11a38 K11a40

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,6,15,5 X2837 X18,9,19,10 X20,11,21,12 X6,14,7,13 X22,16,1,15 X12,17,13,18 X10,19,11,20 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 18 20 6 22 12 10 16

Alexander Polynomial:

- 3t-3 + 12t-2 - 22t-1 + 27 - 22t + 12t2 - 3t3

Conway Polynomial:

1 - z2 - 6z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{101, 0}

Jones Polynomial:

q-4 - 3q-3 + 7q-2 - 11q-1 + 14 - 16q + 16q2 - 13q3 + 10q4 - 6q5 + 3q6 - q7

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

{...}

A2 (sl(3)) Invariant:

q-12 - q-10 + q-8 + 2q-6 - 2q-4 + 3q-2 - 2 - q2 - 3q6 + 3q8 - q10 + 2q12 + 3q14 - q16 + q18 - q20 - q22

HOMFLY-PT Polynomial:

- 2a-6 - a-6z2 + 6a-4 + 9a-4z2 + 3a-4z4 - 5a-2 - 11a-2z2 - 8a-2z4 - 2a-2z6 + 1 - 2z4 - z6 + a2 + 2a2z2 + a2z4

Kauffman Polynomial:

- 2a-7z + 5a-7z3 - 4a-7z5 + a-7z7 + 2a-6 - 7a-6z2 + 15a-6z4 - 12a-6z6 + 3a-6z8 - 4a-5z + 12a-5z3 - 3a-5z5 - 7a-5z7 + 3a-5z9 + 6a-4 - 24a-4z2 + 43a-4z4 - 34a-4z6 + 7a-4z8 + a-4z10 - 4a-3z + 10a-3z3 - 2a-3z5 - 14a-3z7 + 7a-3z9 + 5a-2 - 22a-2z2 + 37a-2z4 - 36a-2z6 + 11a-2z8 + a-2z10 - 5a-1z + 15a-1z3 - 18a-1z5 + 2a-1z7 + 4a-1z9 + 1 + z2 + z4 - 8z6 + 7z8 - 3az + 10az3 - 12az5 + 8az7 - a2 + 5a2z2 - 7a2z4 + 6a2z6 - 2a3z3 + 3a3z5 - a4z2 + a4z4

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

{-1, 2}

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 1139. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 15                       1 j = 13                     2   j = 11                   4 1   j = 9                 6 2     j = 7               7 4       j = 5             9 6         j = 3           7 7           j = 1         7 9             j = -1       5 8               j = -3     2 6                 j = -5   1 5                   j = -7   2                     j = -9 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, 39]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 39]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[2, 8, 3, 7], > X[18, 9, 19, 10], X[20, 11, 21, 12], X[6, 14, 7, 13], X[22, 16, 1, 15], > X[12, 17, 13, 18], X[10, 19, 11, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 39]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -9, 7, -3, 8, -11, 9, -5, 10, > -6, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 39]]
Out[5]=	DTCode[4, 8, 14, 2, 18, 20, 6, 22, 12, 10, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 39]][t]
Out[6]=	3 12 22 2 3 27 - -- + -- - -- - 22 t + 12 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 39]][z]
Out[7]=	2 4 6 1 - z - 6 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 39]}
In[9]:=	{KnotDet[Knot[11, Alternating, 39]], KnotSignature[Knot[11, Alternating, 39]]}
Out[9]=	{101, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 39]][q]
Out[10]=	-4 3 7 11 2 3 4 5 6 7 14 + q - -- + -- - -- - 16 q + 16 q - 13 q + 10 q - 6 q + 3 q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 39]}
In[12]:=	A2Invariant[Knot[11, Alternating, 39]][q]
Out[12]=	-12 -10 -8 2 2 3 2 6 8 10 12 -2 + q - q + q + -- - -- + -- - q - 3 q + 3 q - q + 2 q + 6 4 2 q q q 14 16 18 20 22 > 3 q - q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 39]][a, z]
Out[13]=	2 2 2 4 4 2 6 5 2 z 9 z 11 z 2 2 4 3 z 8 z 1 - -- + -- - -- + a - -- + ---- - ----- + 2 a z - 2 z + ---- - ---- + 6 4 2 6 4 2 4 2 a a a a a a a a 6 2 4 6 2 z > a z - z - ---- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 39]][a, z]
Out[14]=	2 2 2 6 5 2 2 z 4 z 4 z 5 z 2 7 z 24 z 1 + -- + -- + -- - a - --- - --- - --- - --- - 3 a z + z - ---- - ----- - 6 4 2 7 5 3 a 6 4 a a a a a a a a 2 3 3 3 3 22 z 2 2 4 2 5 z 12 z 10 z 15 z 3 > ----- + 5 a z - a z + ---- + ----- + ----- + ----- + 10 a z - 2 7 5 3 a a a a a 4 4 4 5 5 3 3 4 15 z 43 z 37 z 2 4 4 4 4 z 3 z > 2 a z + z + ----- + ----- + ----- - 7 a z + a z - ---- - ---- - 6 4 2 7 5 a a a a a 5 5 6 6 6 2 z 18 z 5 3 5 6 12 z 34 z 36 z 2 6 > ---- - ----- - 12 a z + 3 a z - 8 z - ----- - ----- - ----- + 6 a z + 3 a 6 4 2 a a a a 7 7 7 7 8 8 8 9 z 7 z 14 z 2 z 7 8 3 z 7 z 11 z 3 z > -- - ---- - ----- + ---- + 8 a z + 7 z + ---- + ---- + ----- + ---- + 7 5 3 a 6 4 2 5 a a a a a a a 9 9 10 10 7 z 4 z z z > ---- + ---- + --- + --- 3 a 4 2 a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 39]], Vassiliev[3][Knot[11, Alternating, 39]]}
Out[15]=	{-1, 2}
In[16]:=	Kh[Knot[11, Alternating, 39]][q, t]
Out[16]=	8 1 2 1 5 2 6 5 3 - + 7 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 q t + 7 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 7 q t + 9 q t + 6 q t + 7 q t + 4 q t + 6 q t + 2 q t + 11 5 11 6 13 6 15 7 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a39

K11a38 K11a40