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

K11a327 K11a329

Knotscape

This page is passe. Go here instead!    The Knot K11a328 Visit K11a328's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 18 16 22 4 20 8 2 10 14

Alexander Polynomial:

- 3t-3 + 16t-2 - 34t-1 + 43 - 34t + 16t2 - 3t3

Conway Polynomial:

1 + 3z2 - 2z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{149, 4}

Jones Polynomial:

1 - 4q + 10q2 - 15q3 + 21q4 - 24q5 + 24q6 - 21q7 + 15q8 - 9q9 + 4q10 - q11

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

{...}

A2 (sl(3)) Invariant:

1 - 2q2 + q4 + 3q6 - 3q8 + 6q10 - q12 + 2q16 - 5q18 + 3q20 - 4q22 + q24 + 3q26 - 3q28 + 2q30 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 3a-8 + 6a-8z2 + 3a-8z4 - 5a-6 - 8a-6z2 - 6a-6z4 - 2a-6z6 + 4a-4 + 5a-4z2 - a-4z6 + a-2z2 + a-2z4

Kauffman Polynomial:

- a-13z3 + a-13z5 + a-12z2 - 5a-12z4 + 4a-12z6 - 2a-11z + 6a-11z3 - 12a-11z5 + 8a-11z7 + a-10 - 4a-10z2 + 9a-10z4 - 15a-10z6 + 10a-10z8 - 3a-9z + 12a-9z3 - 8a-9z5 - 6a-9z7 + 8a-9z9 + 3a-8 - 17a-8z2 + 43a-8z4 - 42a-8z6 + 13a-8z8 + 3a-8z10 - 3a-7z + 2a-7z3 + 19a-7z5 - 33a-7z7 + 16a-7z9 + 5a-6 - 23a-6z2 + 46a-6z4 - 43a-6z6 + 11a-6z8 + 3a-6z10 - 2a-5z + 6a-5z5 - 15a-5z7 + 8a-5z9 + 4a-4 - 10a-4z2 + 15a-4z4 - 19a-4z6 + 8a-4z8 + 3a-3z3 - 8a-3z5 + 4a-3z7 + a-2z2 - 2a-2z4 + a-2z6

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

{3, 6}

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 11328. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                       1 j = 21                     3   j = 19                   6 1   j = 17                 9 3     j = 15               12 6       j = 13             12 9         j = 11           12 12           j = 9         9 12             j = 7       6 12               j = 5     4 9                 j = 3   1 7                   j = 1   3                     j = -1 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, 328]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 328]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[18, 6, 19, 5], X[16, 7, 17, 8], > X[22, 10, 1, 9], X[4, 12, 5, 11], X[20, 14, 21, 13], X[8, 15, 9, 16], > X[2, 18, 3, 17], X[10, 20, 11, 19], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 328]]
Out[4]=	GaussCode[1, -9, 2, -6, 3, -1, 4, -8, 5, -10, 6, -2, 7, -11, 8, -4, 9, -3, 10, > -7, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 328]]
Out[5]=	DTCode[6, 12, 18, 16, 22, 4, 20, 8, 2, 10, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 328]][t]
Out[6]=	3 16 34 2 3 43 - -- + -- - -- - 34 t + 16 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 328]][z]
Out[7]=	2 4 6 1 + 3 z - 2 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 328]}
In[9]:=	{KnotDet[Knot[11, Alternating, 328]], KnotSignature[Knot[11, Alternating, 328]]}
Out[9]=	{149, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 328]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 4 q + 10 q - 15 q + 21 q - 24 q + 24 q - 21 q + 15 q - 9 q + 10 11 > 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 328]}
In[12]:=	A2Invariant[Knot[11, Alternating, 328]][q]
Out[12]=	2 4 6 8 10 12 16 18 20 22 1 - 2 q + q + 3 q - 3 q + 6 q - q + 2 q - 5 q + 3 q - 4 q + 24 26 28 30 34 > q + 3 q - 3 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 328]][a, z]
Out[13]=	2 2 2 2 2 4 4 4 -10 3 5 4 z 6 z 8 z 5 z z 3 z 6 z z -a + -- - -- + -- - --- + ---- - ---- + ---- + -- + ---- - ---- + -- - 8 6 4 10 8 6 4 2 8 6 2 a a a a a a a a a a a 6 6 2 z z > ---- - -- 6 4 a a
In[14]:=	Kauffman[Knot[11, Alternating, 328]][a, z]
Out[14]=	2 2 2 2 -10 3 5 4 2 z 3 z 3 z 2 z z 4 z 17 z 23 z a + -- + -- + -- - --- - --- - --- - --- + --- - ---- - ----- - ----- - 8 6 4 11 9 7 5 12 10 8 6 a a a a a a a a a a a 2 2 3 3 3 3 3 4 4 4 10 z z z 6 z 12 z 2 z 3 z 5 z 9 z 43 z > ----- + -- - --- + ---- + ----- + ---- + ---- - ---- + ---- + ----- + 4 2 13 11 9 7 3 12 10 8 a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 46 z 15 z 2 z z 12 z 8 z 19 z 6 z 8 z 4 z > ----- + ----- - ---- + --- - ----- - ---- + ----- + ---- - ---- + ---- - 6 4 2 13 11 9 7 5 3 12 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 15 z 42 z 43 z 19 z z 8 z 6 z 33 z 15 z 4 z > ----- - ----- - ----- - ----- + -- + ---- - ---- - ----- - ----- + ---- + 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 10 z 13 z 11 z 8 z 8 z 16 z 8 z 3 z 3 z > ----- + ----- + ----- + ---- + ---- + ----- + ---- + ----- + ----- 10 8 6 4 9 7 5 8 6 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 328]], Vassiliev[3][Knot[11, Alternating, 328]]}
Out[15]=	{3, 6}
In[16]:=	Kh[Knot[11, Alternating, 328]][q, t]
Out[16]=	3 3 5 1 3 q q 5 7 7 2 9 2 7 q + 4 q + ---- + --- + -- + 9 q t + 6 q t + 12 q t + 9 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 > 12 q t + 12 q t + 12 q t + 12 q t + 9 q t + 12 q t + 15 6 17 6 17 7 19 7 19 8 21 8 23 9 > 6 q t + 9 q t + 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a328

K11a327 K11a329