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

K11a39 K11a41

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 18 20 16 6 22 10 12

Alexander Polynomial:

t-4 - 5t-3 + 12t-2 - 17t-1 + 19 - 17t + 12t2 - 5t3 + t4

Conway Polynomial:

1 + 2z2 + 2z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{K11a330, ...}

Determinant and Signature:

{89, 4}

Jones Polynomial:

- q-1 + 3 - 5q + 9q2 - 11q3 + 14q4 - 14q5 + 12q6 - 10q7 + 6q8 - 3q9 + q10

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

{...}

A2 (sl(3)) Invariant:

- q-2 + 1 - q2 + q4 + 2q6 + 5q10 - q12 + 2q14 - q16 - 3q18 - 3q22 + q24 + q30

HOMFLY-PT Polynomial:

2a-8 + 3a-8z2 + a-8z4 - 7a-6 - 13a-6z2 - 9a-6z4 - 2a-6z6 + 7a-4 + 16a-4z2 + 14a-4z4 + 6a-4z6 + a-4z8 - a-2 - 4a-2z2 - 4a-2z4 - a-2z6

Kauffman Polynomial:

- a-12z2 + a-12z4 + a-11z - 3a-11z3 + 3a-11z5 + 2a-10z2 - 5a-10z4 + 5a-10z6 - a-9z + 4a-9z3 - 7a-9z5 + 6a-9z7 + 2a-8 - a-8z2 - a-8z4 - 4a-8z6 + 5a-8z8 - 8a-7z + 17a-7z3 - 16a-7z5 + 2a-7z7 + 3a-7z9 + 7a-6 - 21a-6z2 + 30a-6z4 - 26a-6z6 + 7a-6z8 + a-6z10 - 8a-5z + 14a-5z3 - 14a-5z7 + 6a-5z9 + 7a-4 - 25a-4z2 + 42a-4z4 - 30a-4z6 + 5a-4z8 + a-4z10 - 3a-3z + 8a-3z3 + 2a-3z5 - 9a-3z7 + 3a-3z9 + a-2 - 8a-2z2 + 17a-2z4 - 13a-2z6 + 3a-2z8 - a-1z + 4a-1z3 - 4a-1z5 + a-1z7

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

{2, 1}

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 1140. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 21                       1 j = 19                     2   j = 17                   4 1   j = 15                 6 2     j = 13               6 4       j = 11             8 6         j = 9           6 6           j = 7         5 8             j = 5       4 6               j = 3     2 6                 j = 1   1 3                   j = -1   2                     j = -3 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, 40]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 40]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 5, 15, 6], X[2, 8, 3, 7], > X[18, 10, 19, 9], X[20, 12, 21, 11], X[16, 13, 17, 14], X[6, 15, 7, 16], > X[22, 18, 1, 17], X[10, 20, 11, 19], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 40]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -10, 6, -11, 7, -3, 8, -7, 9, -5, 10, > -6, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 40]]
Out[5]=	DTCode[4, 8, 14, 2, 18, 20, 16, 6, 22, 10, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 40]][t]
Out[6]=	-4 5 12 17 2 3 4 19 + t - -- + -- - -- - 17 t + 12 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 40]][z]
Out[7]=	2 4 6 8 1 + 2 z + 2 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 40], Knot[11, Alternating, 330]}
In[9]:=	{KnotDet[Knot[11, Alternating, 40]], KnotSignature[Knot[11, Alternating, 40]]}
Out[9]=	{89, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 40]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 10 3 - - - 5 q + 9 q - 11 q + 14 q - 14 q + 12 q - 10 q + 6 q - 3 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 40]}
In[12]:=	A2Invariant[Knot[11, Alternating, 40]][q]
Out[12]=	-2 2 4 6 10 12 14 16 18 22 24 30 1 - q - q + q + 2 q + 5 q - q + 2 q - q - 3 q - 3 q + q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 40]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 2 7 7 -2 3 z 13 z 16 z 4 z z 9 z 14 z 4 z -- - -- + -- - a + ---- - ----- + ----- - ---- + -- - ---- + ----- - ---- - 8 6 4 8 6 4 2 8 6 4 2 a a a a a a a a a a a 6 6 6 8 2 z 6 z z z > ---- + ---- - -- + -- 6 4 2 4 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 40]][a, z]
Out[14]=	2 2 2 2 2 7 7 -2 z z 8 z 8 z 3 z z z 2 z z 21 z -- + -- + -- + a + --- - -- - --- - --- - --- - - - --- + ---- - -- - ----- - 8 6 4 11 9 7 5 3 a 12 10 8 6 a a a a a a a a a a a a 2 2 3 3 3 3 3 3 4 4 25 z 8 z 3 z 4 z 17 z 14 z 8 z 4 z z 5 z > ----- - ---- - ---- + ---- + ----- + ----- + ---- + ---- + --- - ---- - 4 2 11 9 7 5 3 a 12 10 a a a a a a a a a 4 4 4 4 5 5 5 5 5 6 z 30 z 42 z 17 z 3 z 7 z 16 z 2 z 4 z 5 z > -- + ----- + ----- + ----- + ---- - ---- - ----- + ---- - ---- + ---- - 8 6 4 2 11 9 7 3 a 10 a a a a a a a a a 6 6 6 6 7 7 7 7 7 8 4 z 26 z 30 z 13 z 6 z 2 z 14 z 9 z z 5 z > ---- - ----- - ----- - ----- + ---- + ---- - ----- - ---- + -- + ---- + 8 6 4 2 9 7 5 3 a 8 a a a a a a a a a 8 8 8 9 9 9 10 10 7 z 5 z 3 z 3 z 6 z 3 z z z > ---- + ---- + ---- + ---- + ---- + ---- + --- + --- 6 4 2 7 5 3 6 4 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 40]], Vassiliev[3][Knot[11, Alternating, 40]]}
Out[15]=	{2, 1}
In[16]:=	Kh[Knot[11, Alternating, 40]][q, t]
Out[16]=	3 3 5 1 2 q 3 q 2 q 5 7 7 2 6 q + 4 q + ----- + ---- + -- + --- + ---- + 6 q t + 5 q t + 8 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 15 5 > 6 q t + 6 q t + 8 q t + 6 q t + 6 q t + 4 q t + 6 q t + 15 6 17 6 17 7 19 7 21 8 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a40

K11a39 K11a41