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

K11a163 K11a165

Knotscape

This page is passe. Go here instead!    The Knot K11a164 Visit K11a164's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 14 16 2 20 22 12 6 8

Alexander Polynomial:

t-4 - 7t-3 + 20t-2 - 35t-1 + 43 - 35t + 20t2 - 7t3 + t4

Conway Polynomial:

1 - 2z2 - 2z4 + z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{169, 0}

Jones Polynomial:

q-6 - 4q-5 + 10q-4 - 17q-3 + 23q-2 - 27q-1 + 28 - 24q + 18q2 - 11q3 + 5q4 - q5

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

{...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

a-2 - 2a-2z4 - a-2z6 + z2 + 5z4 + 4z6 + z8 - a2 - 5a2z2 - 6a2z4 - 2a2z6 + a4 + 2a4z2 + a4z4

Kauffman Polynomial:

a-5z5 - 4a-4z4 + 5a-4z6 + 4a-3z3 - 14a-3z5 + 11a-3z7 - a-2 + a-2z2 + 6a-2z4 - 19a-2z6 + 14a-2z8 - a-1z + 12a-1z3 - 19a-1z5 - 2a-1z7 + 10a-1z9 - 5z2 + 36z4 - 55z6 + 22z8 + 3z10 - az + 8az3 + az5 - 27az7 + 18az9 + a2 - 13a2z2 + 41a2z4 - 49a2z6 + 16a2z8 + 3a2z10 - a3z + 5a3z3 - 3a3z5 - 10a3z7 + 8a3z9 + a4 - 6a4z2 + 13a4z4 - 17a4z6 + 8a4z8 - a5z + 5a5z3 - 8a5z5 + 4a5z7 + a6z2 - 2a6z4 + a6z6

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=0 is the signature of 11164. 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               13 7       j = 1             15 11         j = -1           13 14           j = -3         10 14             j = -5       7 13               j = -7     3 10                 j = -9   1 7                   j = -11   3                     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, 164]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 164]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 6, 19, 5], X[14, 8, 15, 7], > X[16, 9, 17, 10], X[2, 11, 3, 12], X[20, 13, 21, 14], X[22, 16, 1, 15], > X[12, 17, 13, 18], X[6, 20, 7, 19], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 164]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -11, 5, -2, 6, -9, 7, -4, 8, -5, 9, -3, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 164]]
Out[5]=	DTCode[4, 10, 18, 14, 16, 2, 20, 22, 12, 6, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 164]][t]
Out[6]=	-4 7 20 35 2 3 4 43 + t - -- + -- - -- - 35 t + 20 t - 7 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 164]][z]
Out[7]=	2 4 6 8 1 - 2 z - 2 z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 164]}
In[9]:=	{KnotDet[Knot[11, Alternating, 164]], KnotSignature[Knot[11, Alternating, 164]]}
Out[9]=	{169, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 164]][q]
Out[10]=	-6 4 10 17 23 27 2 3 4 5 28 + q - -- + -- - -- + -- - -- - 24 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, 164]}
In[12]:=	A2Invariant[Knot[11, Alternating, 164]][q]
Out[12]=	-18 -16 3 4 4 2 2 4 2 4 6 8 -5 + q - q + --- - --- + -- - -- - -- + -- + 6 q - 4 q + q + 3 q - 12 10 8 6 4 2 q q q q q q 10 12 14 > 3 q + 3 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 164]][a, z]
Out[13]=	4 -2 2 4 2 2 2 4 2 4 2 z 2 4 4 4 6 a - a + a + z - 5 a z + 2 a z + 5 z - ---- - 6 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, 164]][a, z]
Out[14]=	2 -2 2 4 z 3 5 2 z 2 2 4 2 -a + a + a - - - a z - a z - a z - 5 z + -- - 13 a z - 6 a z + a 2 a 3 3 4 4 6 2 4 z 12 z 3 3 3 5 3 4 4 z 6 z > a z + ---- + ----- + 8 a z + 5 a z + 5 a z + 36 z - ---- + ---- + 3 a 4 2 a a a 5 5 5 2 4 4 4 6 4 z 14 z 19 z 5 3 5 > 41 a z + 13 a z - 2 a z + -- - ----- - ----- + a z - 3 a z - 5 3 a a a 6 6 7 5 5 6 5 z 19 z 2 6 4 6 6 6 11 z > 8 a z - 55 z + ---- - ----- - 49 a z - 17 a z + a z + ----- - 4 2 3 a a a 7 8 2 z 7 3 7 5 7 8 14 z 2 8 4 8 > ---- - 27 a z - 10 a z + 4 a z + 22 z + ----- + 16 a z + 8 a z + a 2 a 9 10 z 9 3 9 10 2 10 > ----- + 18 a z + 8 a z + 3 z + 3 a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 164]], Vassiliev[3][Knot[11, Alternating, 164]]}
Out[15]=	{-2, 1}
In[16]:=	Kh[Knot[11, Alternating, 164]][q, t]
Out[16]=	14 1 3 1 7 3 10 7 13 -- + 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 10 14 13 3 3 2 5 2 5 3 > ----- + ---- + --- + 11 q t + 13 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 K11a164

K11a163 K11a165