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

K11a18 K11a20

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 12 2 16 18 6 20 22 14 10

Alexander Polynomial:

- t-4 + 6t-3 - 18t-2 + 33t-1 - 39 + 33t - 18t2 + 6t3 - t4

Conway Polynomial:

1 - z2 - 2z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a25, K11a281, ...}

Determinant and Signature:

{155, 2}

Jones Polynomial:

q-3 - 4q-2 + 10q-1 - 16 + 22q - 25q2 + 25q3 - 22q4 + 16q5 - 9q6 + 4q7 - q8

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

{K11a25, ...}

A2 (sl(3)) Invariant:

q-8 - 2q-6 + 4q-4 - q-2 + 4q2 - 6q4 + 4q6 - 4q8 + q10 + 2q12 - 3q14 + 5q16 - 2q18 + q22 - q24

HOMFLY-PT Polynomial:

- a-6 - 2a-6z2 - a-6z4 + 4a-4 + 9a-4z2 + 7a-4z4 + 2a-4z6 - 5a-2 - 12a-2z2 - 11a-2z4 - 5a-2z6 - a-2z8 + 3 + 4z2 + 3z4 + z6

Kauffman Polynomial:

- a-9z3 + a-9z5 + 2a-8z2 - 5a-8z4 + 4a-8z6 - 2a-7z + 7a-7z3 - 11a-7z5 + 8a-7z7 + a-6 - a-6z2 + 5a-6z4 - 12a-6z6 + 10a-6z8 - 6a-5z + 23a-5z3 - 26a-5z5 + 4a-5z7 + 7a-5z9 + 4a-4 - 18a-4z2 + 43a-4z4 - 52a-4z6 + 21a-4z8 + 2a-4z10 - 6a-3z + 21a-3z3 - 17a-3z5 - 14a-3z7 + 14a-3z9 + 5a-2 - 24a-2z2 + 50a-2z4 - 55a-2z6 + 19a-2z8 + 2a-2z10 - 3a-1z + 11a-1z3 - 11a-1z5 - 6a-1z7 + 7a-1z9 + 3 - 8z2 + 15z4 - 18z6 + 8z8 - az + 5az3 - 8az5 + 4az7 + a2z2 - 2a2z4 + a2z6

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

{-1, 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=2 is the signature of 1119. 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 = 17                       1 j = 15                     3   j = 13                   6 1   j = 11                 10 3     j = 9               12 6       j = 7             13 10         j = 5           12 12           j = 3         10 13             j = 1       7 13               j = -1     3 9                 j = -3   1 7                   j = -5   3                     j = -7 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, 19]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 19]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[16, 9, 17, 10], X[18, 12, 19, 11], X[6, 13, 7, 14], X[20, 16, 21, 15], > X[22, 17, 1, 18], X[14, 20, 15, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 19]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -11, 6, -3, 7, -10, 8, -5, 9, -6, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 19]]
Out[5]=	DTCode[4, 8, 12, 2, 16, 18, 6, 20, 22, 14, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 19]][t]
Out[6]=	-4 6 18 33 2 3 4 -39 - t + -- - -- + -- + 33 t - 18 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 19]][z]
Out[7]=	2 4 6 8 1 - z - 2 z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 19], Knot[11, Alternating, 25], > Knot[11, Alternating, 281]}
In[9]:=	{KnotDet[Knot[11, Alternating, 19]], KnotSignature[Knot[11, Alternating, 19]]}
Out[9]=	{155, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 19]][q]
Out[10]=	-3 4 10 2 3 4 5 6 7 8 -16 + q - -- + -- + 22 q - 25 q + 25 q - 22 q + 16 q - 9 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 19], Knot[11, Alternating, 25]}
In[12]:=	A2Invariant[Knot[11, Alternating, 19]][q]
Out[12]=	-8 2 4 -2 2 4 6 8 10 12 14 16 q - -- + -- - q + 4 q - 6 q + 4 q - 4 q + q + 2 q - 3 q + 5 q - 6 4 q q 18 22 24 > 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 19]][a, z]
Out[13]=	2 2 2 4 4 4 -6 4 5 2 2 z 9 z 12 z 4 z 7 z 11 z 3 - a + -- - -- + 4 z - ---- + ---- - ----- + 3 z - -- + ---- - ----- + 4 2 6 4 2 6 4 2 a a a a a a a a 6 6 8 6 2 z 5 z z > z + ---- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 19]][a, z]
Out[14]=	2 2 2 -6 4 5 2 z 6 z 6 z 3 z 2 2 z z 18 z 3 + a + -- + -- - --- - --- - --- - --- - a z - 8 z + ---- - -- - ----- - 4 2 7 5 3 a 8 6 4 a a a a a a a a 2 3 3 3 3 3 4 24 z 2 2 z 7 z 23 z 21 z 11 z 3 4 5 z > ----- + a z - -- + ---- + ----- + ----- + ----- + 5 a z + 15 z - ---- + 2 9 7 5 3 a 8 a a a a a a 4 4 4 5 5 5 5 5 5 z 43 z 50 z 2 4 z 11 z 26 z 17 z 11 z > ---- + ----- + ----- - 2 a z + -- - ----- - ----- - ----- - ----- - 6 4 2 9 7 5 3 a a a a a a a a 6 6 6 6 7 7 5 6 4 z 12 z 52 z 55 z 2 6 8 z 4 z > 8 a z - 18 z + ---- - ----- - ----- - ----- + a z + ---- + ---- - 8 6 4 2 7 5 a a a a a a 7 7 8 8 8 9 9 14 z 6 z 7 8 10 z 21 z 19 z 7 z 14 z > ----- - ---- + 4 a z + 8 z + ----- + ----- + ----- + ---- + ----- + 3 a 6 4 2 5 3 a a a a a a 9 10 10 7 z 2 z 2 z > ---- + ----- + ----- a 4 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 19]], Vassiliev[3][Knot[11, Alternating, 19]]}
Out[15]=	{-1, 0}
In[16]:=	Kh[Knot[11, Alternating, 19]][q, t]
Out[16]=	3 1 3 1 7 3 9 7 q 3 13 q + 10 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 13 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 12 q t + 12 q t + 13 q t + 10 q t + 12 q t + 6 q t + 10 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a19

K11a18 K11a20