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

K11a100 K11a102

Knotscape

This page is passe. Go here instead!    The Knot K11a101 Visit K11a101's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 12 2 20 18 22 6 8 16

Alexander Polynomial:

3t-3 - 16t-2 + 39t-1 - 51 + 39t - 16t2 + 3t3

Conway Polynomial:

1 + 2z2 + 2z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{167, 2}

Jones Polynomial:

- q-2 + 4q-1 - 9 + 17q - 23q2 + 27q3 - 27q4 + 24q5 - 18q6 + 11q7 - 5q8 + q9

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

{...}

A2 (sl(3)) Invariant:

- q-6 + 2q-4 - q-2 - 2 + 6q2 - 4q4 + 3q6 + 2q8 - 3q10 + 4q12 - 5q14 + 4q16 - 3q20 + 4q22 - 3q24 - q26 + q28

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-10z4 + a-10z6 + 4a-9z3 - 9a-9z5 + 5a-9z7 - 2a-8z2 + 12a-8z4 - 21a-8z6 + 10a-8z8 - 3a-7z + 13a-7z3 - 13a-7z5 - 9a-7z7 + 9a-7z9 + a-6 - 12a-6z2 + 43a-6z4 - 57a-6z6 + 19a-6z8 + 3a-6z10 - 5a-5z + 16a-5z3 - 5a-5z5 - 26a-5z7 + 18a-5z9 + a-4 - 13a-4z2 + 44a-4z4 - 54a-4z6 + 20a-4z8 + 3a-4z10 - 3a-3z + 13a-3z3 - 12a-3z5 - 4a-3z7 + 9a-3z9 - a-2 - a-2z2 + 9a-2z4 - 15a-2z6 + 11a-2z8 - a-1z + 5a-1z3 - 10a-1z5 + 8a-1z7 + 2z2 - 5z4 + 4z6 - az3 + az5

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

{2, 3}

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 11101. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                       1 j = 17                     4   j = 15                   7 1   j = 13                 11 4     j = 11               13 7       j = 9             14 11         j = 7           13 13           j = 5         10 14             j = 3       7 13               j = 1     3 11                 j = -1   1 6                   j = -3   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, 101]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 101]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[12, 7, 13, 8], > X[2, 10, 3, 9], X[20, 11, 21, 12], X[18, 14, 19, 13], X[22, 16, 1, 15], > X[6, 18, 7, 17], X[8, 19, 9, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 101]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -10, 5, -2, 6, -4, 7, -3, 8, -11, 9, -7, 10, > -6, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 101]]
Out[5]=	DTCode[4, 10, 14, 12, 2, 20, 18, 22, 6, 8, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 101]][t]
Out[6]=	3 16 39 2 3 -51 + -- - -- + -- + 39 t - 16 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 101]][z]
Out[7]=	2 4 6 1 + 2 z + 2 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 101]}
In[9]:=	{KnotDet[Knot[11, Alternating, 101]], KnotSignature[Knot[11, Alternating, 101]]}
Out[9]=	{167, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 101]][q]
Out[10]=	-2 4 2 3 4 5 6 7 8 9 -9 - q + - + 17 q - 23 q + 27 q - 27 q + 24 q - 18 q + 11 q - 5 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 101]}
In[12]:=	A2Invariant[Knot[11, Alternating, 101]][q]
Out[12]=	-6 2 -2 2 4 6 8 10 12 14 -2 - q + -- - q + 6 q - 4 q + 3 q + 2 q - 3 q + 4 q - 5 q + 4 q 16 20 22 24 26 28 > 4 q - 3 q + 4 q - 3 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 101]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -6 -4 -2 2 z 4 z 5 z z 4 3 z 5 z z 2 z -a + a + a - z + -- - ---- + ---- + -- - z - ---- + ---- + -- + ---- + 8 6 4 2 6 4 2 4 a a a a a a a a 6 z > -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 101]][a, z]
Out[14]=	2 2 2 2 -6 -4 -2 3 z 5 z 3 z z 2 2 z 12 z 13 z z a + a - a - --- - --- - --- - - + 2 z - ---- - ----- - ----- - -- + 7 5 3 a 8 6 4 2 a a a a a a a 3 3 3 3 3 4 4 4 4 z 13 z 16 z 13 z 5 z 3 4 z 12 z 43 z > ---- + ----- + ----- + ----- + ---- - a z - 5 z - --- + ----- + ----- + 9 7 5 3 a 10 8 6 a a a a a a a 4 4 5 5 5 5 5 6 44 z 9 z 9 z 13 z 5 z 12 z 10 z 5 6 z > ----- + ---- - ---- - ----- - ---- - ----- - ----- + a z + 4 z + --- - 4 2 9 7 5 3 a 10 a a a a a a a 6 6 6 6 7 7 7 7 7 8 21 z 57 z 54 z 15 z 5 z 9 z 26 z 4 z 8 z 10 z > ----- - ----- - ----- - ----- + ---- - ---- - ----- - ---- + ---- + ----- + 8 6 4 2 9 7 5 3 a 8 a a a a a a a a a 8 8 8 9 9 9 10 10 19 z 20 z 11 z 9 z 18 z 9 z 3 z 3 z > ----- + ----- + ----- + ---- + ----- + ---- + ----- + ----- 6 4 2 7 5 3 6 4 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 101]], Vassiliev[3][Knot[11, Alternating, 101]]}
Out[15]=	{2, 3}
In[16]:=	Kh[Knot[11, Alternating, 101]][q, t]
Out[16]=	3 1 3 1 6 3 q 3 5 5 2 11 q + 7 q + ----- + ----- + ---- + --- + --- + 13 q t + 10 q t + 14 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 > 13 q t + 13 q t + 14 q t + 11 q t + 13 q t + 7 q t + 13 5 13 6 15 6 15 7 17 7 19 8 > 11 q t + 4 q t + 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a101

K11a100 K11a102