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

K11a139 K11a141

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 18 2 22 20 8 6 14 12

Alexander Polynomial:

- 2t-3 + 8t-2 - 14t-1 + 17 - 14t + 8t2 - 2t3

Conway Polynomial:

1 - 4z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{1025, 1056, ...}

Determinant and Signature:

{65, 4}

Jones Polynomial:

q-2 - 2q-1 + 4 - 6q + 8q2 - 9q3 + 10q4 - 9q5 + 7q6 - 5q7 + 3q8 - q9

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

{K11a9, ...}

A2 (sl(3)) Invariant:

q-6 + q-2 + 1 - q2 + q4 - q6 - q8 + q10 - q12 + 2q14 + q20 - q22 + q24 - q28

HOMFLY-PT Polynomial:

- a-8 - a-8z2 + 2a-6 + 5a-6z2 + 2a-6z4 - 2a-4z2 - 3a-4z4 - a-4z6 - 2a-2 - 5a-2z2 - 4a-2z4 - a-2z6 + 2 + 3z2 + z4

Kauffman Polynomial:

a-11z3 - a-10z2 + 3a-10z4 + a-9z - 4a-9z3 + 5a-9z5 - a-8 + 2a-8z2 - 7a-8z4 + 6a-8z6 - 9a-7z5 + 6a-7z7 - 2a-6 + 6a-6z2 - 3a-6z4 - 9a-6z6 + 5a-6z8 - 2a-5z + 11a-5z3 - 10a-5z5 - 4a-5z7 + 3a-5z9 - 5a-4z2 + 24a-4z4 - 22a-4z6 + 3a-4z8 + a-4z10 - 5a-3z3 + 23a-3z5 - 21a-3z7 + 5a-3z9 + 2a-2 - 17a-2z2 + 29a-2z4 - 13a-2z6 - a-2z8 + a-2z10 + a-1z - 11a-1z3 + 19a-1z5 - 11a-1z7 + 2a-1z9 + 2 - 9z2 + 12z4 - 6z6 + z8

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

{0, 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=4 is the signature of 11140. 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 = 19                       1 j = 17                     2   j = 15                   3 1   j = 13                 4 2     j = 11               5 3       j = 9             5 4         j = 7           4 5           j = 5         4 5             j = 3       3 5               j = 1     1 3                 j = -1   1 3                   j = -3   1                     j = -5 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, 140]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 140]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[16, 6, 17, 5], X[18, 8, 19, 7], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[20, 13, 21, 14], X[8, 16, 9, 15], > X[6, 18, 7, 17], X[14, 19, 15, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 140]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -8, 5, -2, 6, -11, 7, -10, 8, -3, 9, -4, 10, > -7, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 140]]
Out[5]=	DTCode[4, 10, 16, 18, 2, 22, 20, 8, 6, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 140]][t]
Out[6]=	2 8 14 2 3 17 - -- + -- - -- - 14 t + 8 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 140]][z]
Out[7]=	4 6 1 - 4 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 25], Knot[10, 56], Knot[11, Alternating, 140]}
In[9]:=	{KnotDet[Knot[11, Alternating, 140]], KnotSignature[Knot[11, Alternating, 140]]}
Out[9]=	{65, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 140]][q]
Out[10]=	-2 2 2 3 4 5 6 7 8 9 4 + q - - - 6 q + 8 q - 9 q + 10 q - 9 q + 7 q - 5 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 9], Knot[11, Alternating, 140]}
In[12]:=	A2Invariant[Knot[11, Alternating, 140]][q]
Out[12]=	-6 -2 2 4 6 8 10 12 14 20 22 24 28 1 + q + q - q + q - q - q + q - q + 2 q + q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 140]][a, z]
Out[13]=	2 2 2 2 4 4 4 -8 2 2 2 z 5 z 2 z 5 z 4 2 z 3 z 4 z 2 - a + -- - -- + 3 z - -- + ---- - ---- - ---- + z + ---- - ---- - ---- - 6 2 8 6 4 2 6 4 2 a a a a a a a a a 6 6 z z > -- - -- 4 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 140]][a, z]
Out[14]=	2 2 2 2 2 -8 2 2 z 2 z z 2 z 2 z 6 z 5 z 17 z 2 - a - -- + -- + -- - --- + - - 9 z - --- + ---- + ---- - ---- - ----- + 6 2 9 5 a 10 8 6 4 2 a a a a a a a a a 3 3 3 3 3 4 4 4 4 z 4 z 11 z 5 z 11 z 4 3 z 7 z 3 z 24 z > --- - ---- + ----- - ---- - ----- + 12 z + ---- - ---- - ---- + ----- + 11 9 5 3 a 10 8 6 4 a a a a a a a a 4 5 5 5 5 5 6 6 6 29 z 5 z 9 z 10 z 23 z 19 z 6 6 z 9 z 22 z > ----- + ---- - ---- - ----- + ----- + ----- - 6 z + ---- - ---- - ----- - 2 9 7 5 3 a 8 6 4 a a a a a a a a 6 7 7 7 7 8 8 8 9 9 13 z 6 z 4 z 21 z 11 z 8 5 z 3 z z 3 z 5 z > ----- + ---- - ---- - ----- - ----- + z + ---- + ---- - -- + ---- + ---- + 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a 9 10 10 2 z z z > ---- + --- + --- a 4 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 140]], Vassiliev[3][Knot[11, Alternating, 140]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, Alternating, 140]][q, t]
Out[16]=	3 3 5 1 1 1 3 q 3 q 3 q 5 7 5 q + 4 q + ----- + ----- + ---- + ---- + -- + --- + ---- + 5 q t + 4 q t + 5 4 3 3 3 2 2 t t q t q t q t q t t 7 2 9 2 9 3 11 3 11 4 13 4 13 5 > 5 q t + 5 q t + 4 q t + 5 q t + 3 q t + 4 q t + 2 q t + 15 5 15 6 17 6 19 7 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a140

K11a139 K11a141