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

K11a317 K11a319

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X12,4,13,3 X16,6,17,5 X22,8,1,7 X18,10,19,9 X2,12,3,11 X20,14,21,13 X4,16,5,15 X8,18,9,17 X14,20,15,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 16 22 18 2 20 4 8 14 10

Alexander Polynomial:

5t-3 - 16t-2 + 29t-1 - 35 + 29t - 16t2 + 5t3

Conway Polynomial:

1 + 10z2 + 14z4 + 5z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{135, 6}

Jones Polynomial:

q3 - 3q4 + 9q5 - 14q6 + 19q7 - 22q8 + 22q9 - 19q10 + 14q11 - 8q12 + 3q13 - q14

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

{...}

A2 (sl(3)) Invariant:

q10 - 2q12 + 4q14 - q16 + q18 + 4q20 - 4q22 + 4q24 - 3q26 + q28 + 2q30 - 3q32 + 3q34 - 3q36 - q38 + q40 - q42

HOMFLY-PT Polynomial:

- a-12 - 2a-12z2 - a-12z4 - a-10 - 2a-10z2 + a-10z4 + a-10z6 + 2a-8 + 11a-8z2 + 11a-8z4 + 3a-8z6 + a-6 + 3a-6z2 + 3a-6z4 + a-6z6

Kauffman Polynomial:

a-17z - 2a-17z3 + a-17z5 + a-16z2 - 4a-16z4 + 3a-16z6 - 3a-15z + 7a-15z3 - 9a-15z5 + 6a-15z7 - 6a-14z2 + 13a-14z4 - 13a-14z6 + 8a-14z8 - 3a-13z + 11a-13z3 - 9a-13z5 - 2a-13z7 + 6a-13z9 - a-12 + 17a-12z4 - 28a-12z6 + 12a-12z8 + 2a-12z10 - 3a-11z + 13a-11z3 - 8a-11z5 - 13a-11z7 + 11a-11z9 + a-10 - 6a-10z2 + 17a-10z4 - 27a-10z6 + 10a-10z8 + 2a-10z10 - 4a-9z + 14a-9z3 - 15a-9z5 - 2a-9z7 + 5a-9z9 + 2a-8 - 10a-8z2 + 14a-8z4 - 14a-8z6 + 6a-8z8 + 3a-7z3 - 6a-7z5 + 3a-7z7 - a-6 + 3a-6z2 - 3a-6z4 + a-6z6

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

{10, 31}

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=6 is the signature of 11318. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 29                       1 j = 27                     2   j = 25                   6 1   j = 23                 8 2     j = 21               11 6       j = 19             11 8         j = 17           11 11           j = 15         8 11             j = 13       6 11               j = 11     3 8                 j = 9     6                   j = 7 1 3                     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, 318]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 318]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[16, 6, 17, 5], X[22, 8, 1, 7], > X[18, 10, 19, 9], X[2, 12, 3, 11], X[20, 14, 21, 13], X[4, 16, 5, 15], > X[8, 18, 9, 17], X[14, 20, 15, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 318]]
Out[4]=	GaussCode[1, -6, 2, -8, 3, -1, 4, -9, 5, -11, 6, -2, 7, -10, 8, -3, 9, -5, 10, > -7, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 318]]
Out[5]=	DTCode[6, 12, 16, 22, 18, 2, 20, 4, 8, 14, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 318]][t]
Out[6]=	5 16 29 2 3 -35 + -- - -- + -- + 29 t - 16 t + 5 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 318]][z]
Out[7]=	2 4 6 1 + 10 z + 14 z + 5 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 318]}
In[9]:=	{KnotDet[Knot[11, Alternating, 318]], KnotSignature[Knot[11, Alternating, 318]]}
Out[9]=	{135, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 318]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 q - 3 q + 9 q - 14 q + 19 q - 22 q + 22 q - 19 q + 14 q - 8 q + 13 14 > 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 318]}
In[12]:=	A2Invariant[Knot[11, Alternating, 318]][q]
Out[12]=	10 12 14 16 18 20 22 24 26 28 30 q - 2 q + 4 q - q + q + 4 q - 4 q + 4 q - 3 q + q + 2 q - 32 34 36 38 40 42 > 3 q + 3 q - 3 q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 318]][a, z]
Out[13]=	2 2 2 2 4 4 4 -12 -10 2 -6 2 z 2 z 11 z 3 z z z 11 z -a - a + -- + a - ---- - ---- + ----- + ---- - --- + --- + ----- + 8 12 10 8 6 12 10 8 a a a a a a a a 4 6 6 6 3 z z 3 z z > ---- + --- + ---- + -- 6 10 8 6 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 318]][a, z]
Out[14]=	2 2 2 -12 -10 2 -6 z 3 z 3 z 3 z 4 z z 6 z 6 z -a + a + -- - a + --- - --- - --- - --- - --- + --- - ---- - ---- - 8 17 15 13 11 9 16 14 10 a a a a a a a a a 2 2 3 3 3 3 3 3 4 4 10 z 3 z 2 z 7 z 11 z 13 z 14 z 3 z 4 z 13 z > ----- + ---- - ---- + ---- + ----- + ----- + ----- + ---- - ---- + ----- + 8 6 17 15 13 11 9 7 16 14 a a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 17 z 17 z 14 z 3 z z 9 z 9 z 8 z 15 z 6 z > ----- + ----- + ----- - ---- + --- - ---- - ---- - ---- - ----- - ---- + 12 10 8 6 17 15 13 11 9 7 a a a a a a a a a a 6 6 6 6 6 6 7 7 7 7 3 z 13 z 28 z 27 z 14 z z 6 z 2 z 13 z 2 z > ---- - ----- - ----- - ----- - ----- + -- + ---- - ---- - ----- - ---- + 16 14 12 10 8 6 15 13 11 9 a a a a a a a a a a 7 8 8 8 8 9 9 9 10 10 3 z 8 z 12 z 10 z 6 z 6 z 11 z 5 z 2 z 2 z > ---- + ---- + ----- + ----- + ---- + ---- + ----- + ---- + ----- + ----- 7 14 12 10 8 13 11 9 12 10 a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 318]], Vassiliev[3][Knot[11, Alternating, 318]]}
Out[15]=	{10, 31}
In[16]:=	Kh[Knot[11, Alternating, 318]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 q + q + 3 q t + 6 q t + 3 q t + 8 q t + 6 q t + 11 q t + 15 4 15 5 17 5 17 6 19 6 19 7 > 8 q t + 11 q t + 11 q t + 11 q t + 11 q t + 8 q t + 21 7 21 8 23 8 23 9 25 9 25 10 > 11 q t + 6 q t + 8 q t + 2 q t + 6 q t + q t + 27 10 29 11 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a318

K11a317 K11a319