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

K11a72 K11a74

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 14 16 2 18 20 22 6 8

Alexander Polynomial:

t-4 - 7t-3 + 21t-2 - 37t-1 + 45 - 37t + 21t2 - 7t3 + t4

Conway Polynomial:

1 - z4 + z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{177, 0}

Jones Polynomial:

q-6 - 5q-5 + 11q-4 - 18q-3 + 25q-2 - 28q-1 + 29 - 25q + 18q2 - 11q3 + 5q4 - q5

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

{...}

A2 (sl(3)) Invariant:

q-18 - 2q-16 - q-14 + 2q-12 - 4q-10 + 6q-8 + 4q-2 - 6 + 5q2 - 5q4 + q6 + 3q8 - 3q10 + 3q12 - q14

HOMFLY-PT Polynomial:

a-2 - 2a-2z4 - a-2z6 - 2 + 5z4 + 4z6 + z8 + 3a2 - a2z2 - 5a2z4 - 2a2z6 - a4 + a4z2 + a4z4

Kauffman Polynomial:

a-5z5 - 4a-4z4 + 5a-4z6 + 5a-3z3 - 14a-3z5 + 11a-3z7 - a-2 + 6a-2z4 - 18a-2z6 + 14a-2z8 - 2a-1z + 18a-1z3 - 26a-1z5 + a-1z7 + 10a-1z9 - 2 - 3z2 + 38z4 - 61z6 + 25z8 + 3z10 - 4az + 23az3 - 16az5 - 23az7 + 19az9 - 3a2 - 5a2z2 + 42a2z4 - 60a2z6 + 21a2z8 + 3a2z10 - 2a3z + 14a3z3 - 14a3z5 - 8a3z7 + 9a3z9 - a4 - 2a4z2 + 13a4z4 - 21a4z6 + 10a4z8 + 4a5z3 - 9a5z5 + 5a5z7 - a6z4 + a6z6

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

{0, -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=0 is the signature of 1173. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     4   j = 7                   7 1   j = 5                 11 4     j = 3               14 7       j = 1             15 11         j = -1           14 15           j = -3         11 14             j = -5       7 14               j = -7     4 11                 j = -9   1 7                   j = -11   4                     j = -13 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, 73]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 73]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 7, 15, 8], > X[16, 10, 17, 9], X[2, 11, 3, 12], X[18, 13, 19, 14], X[20, 16, 21, 15], > X[22, 17, 1, 18], X[6, 20, 7, 19], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 73]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -11, 5, -2, 6, -3, 7, -4, 8, -5, 9, -7, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 73]]
Out[5]=	DTCode[4, 10, 12, 14, 16, 2, 18, 20, 22, 6, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 73]][t]
Out[6]=	-4 7 21 37 2 3 4 45 + t - -- + -- - -- - 37 t + 21 t - 7 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 73]][z]
Out[7]=	4 6 8 1 - z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 73]}
In[9]:=	{KnotDet[Knot[11, Alternating, 73]], KnotSignature[Knot[11, Alternating, 73]]}
Out[9]=	{177, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 73]][q]
Out[10]=	-6 5 11 18 25 28 2 3 4 5 29 + q - -- + -- - -- + -- - -- - 25 q + 18 q - 11 q + 5 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 73]}
In[12]:=	A2Invariant[Knot[11, Alternating, 73]][q]
Out[12]=	-18 2 -14 2 4 6 4 2 4 6 8 -6 + q - --- - q + --- - --- + -- + -- + 5 q - 5 q + q + 3 q - 16 12 10 8 2 q q q q q 10 12 14 > 3 q + 3 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 73]][a, z]
Out[13]=	4 -2 2 4 2 2 4 2 4 2 z 2 4 4 4 6 -2 + a + 3 a - a - a z + a z + 5 z - ---- - 5 a z + a z + 4 z - 2 a 6 z 2 6 8 > -- - 2 a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 73]][a, z]
Out[14]=	3 -2 2 4 2 z 3 2 2 2 4 2 5 z -2 - a - 3 a - a - --- - 4 a z - 2 a z - 3 z - 5 a z - 2 a z + ---- + a 3 a 3 4 4 18 z 3 3 3 5 3 4 4 z 6 z 2 4 > ----- + 23 a z + 14 a z + 4 a z + 38 z - ---- + ---- + 42 a z + a 4 2 a a 5 5 5 4 4 6 4 z 14 z 26 z 5 3 5 5 5 > 13 a z - a z + -- - ----- - ----- - 16 a z - 14 a z - 9 a z - 5 3 a a a 6 6 7 7 6 5 z 18 z 2 6 4 6 6 6 11 z z 7 > 61 z + ---- - ----- - 60 a z - 21 a z + a z + ----- + -- - 23 a z - 4 2 3 a a a a 8 9 3 7 5 7 8 14 z 2 8 4 8 10 z 9 > 8 a z + 5 a z + 25 z + ----- + 21 a z + 10 a z + ----- + 19 a z + 2 a a 3 9 10 2 10 > 9 a z + 3 z + 3 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 73]], Vassiliev[3][Knot[11, Alternating, 73]]}
Out[15]=	{0, -1}
In[16]:=	Kh[Knot[11, Alternating, 73]][q, t]
Out[16]=	15 1 4 1 7 4 11 7 14 -- + 15 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 11 14 14 3 3 2 5 2 5 3 > ----- + ---- + --- + 11 q t + 14 q t + 7 q t + 11 q t + 4 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a73

K11a72 K11a74