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

K11a138 K11a140

Knotscape

This page is passe. Go here instead!    The Knot K11a139 Visit K11a139's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 18 2 20 22 8 6 14 12

Alexander Polynomial:

- t-4 + 5t-3 - 12t-2 + 20t-1 - 23 + 20t - 12t2 + 5t3 - t4

Conway Polynomial:

1 + z2 - 2z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a57, K11a108, K11a231, ...}

Determinant and Signature:

{99, 2}

Jones Polynomial:

q-3 - 3q-2 + 6q-1 - 10 + 14q - 15q2 + 16q3 - 14q4 + 10q5 - 6q6 + 3q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-8 - q-6 + 2q-4 - q-2 - 1 + 2q2 - 3q4 + 4q6 - q8 + q10 + q12 - 2q14 + 3q16 - q18 - q24

HOMFLY-PT Polynomial:

- 2a-6 - 3a-6z2 - a-6z4 + 5a-4 + 13a-4z2 + 9a-4z4 + 2a-4z6 - 4a-2 - 14a-2z2 - 14a-2z4 - 6a-2z6 - a-2z8 + 2 + 5z2 + 4z4 + z6

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + 2a-8z2 - 6a-8z4 + 3a-8z6 - 3a-7z + 9a-7z3 - 11a-7z5 + 5a-7z7 + 2a-6 - 4a-6z2 + 8a-6z4 - 9a-6z6 + 5a-6z8 - 5a-5z + 17a-5z3 - 12a-5z5 + a-5z7 + 3a-5z9 + 5a-4 - 19a-4z2 + 29a-4z4 - 21a-4z6 + 7a-4z8 + a-4z10 - 3a-3z + 4a-3z3 + 4a-3z5 - 10a-3z7 + 6a-3z9 + 4a-2 - 19a-2z2 + 25a-2z4 - 20a-2z6 + 6a-2z8 + a-2z10 - 2a-1z + 5a-1z3 - 5a-1z5 - 3a-1z7 + 3a-1z9 + 2 - 4z2 + 7z4 - 10z6 + 4z8 - az + 7az3 - 9az5 + 3az7 + 2a2z2 - 3a2z4 + a2z6

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

{1, 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 11139. 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                     2   j = 13                   4 1   j = 11                 6 2     j = 9               8 4       j = 7             8 6         j = 5           7 8           j = 3         7 8             j = 1       4 8               j = -1     2 6                 j = -3   1 4                   j = -5   2                     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, 139]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 139]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[16, 6, 17, 5], X[18, 8, 19, 7], > X[2, 10, 3, 9], X[20, 11, 21, 12], X[22, 13, 1, 14], X[8, 16, 9, 15], > X[6, 18, 7, 17], X[14, 19, 15, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 139]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -8, 5, -2, 6, -11, 7, -10, 8, -3, 9, -4, 10, > -6, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 139]]
Out[5]=	DTCode[4, 10, 16, 18, 2, 20, 22, 8, 6, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 139]][t]
Out[6]=	-4 5 12 20 2 3 4 -23 - t + -- - -- + -- + 20 t - 12 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 139]][z]
Out[7]=	2 4 6 8 1 + z - 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 57], Knot[11, Alternating, 108], > Knot[11, Alternating, 139], Knot[11, Alternating, 231]}
In[9]:=	{KnotDet[Knot[11, Alternating, 139]], KnotSignature[Knot[11, Alternating, 139]]}
Out[9]=	{99, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 139]][q]
Out[10]=	-3 3 6 2 3 4 5 6 7 8 -10 + q - -- + - + 14 q - 15 q + 16 q - 14 q + 10 q - 6 q + 3 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 139]}
In[12]:=	A2Invariant[Knot[11, Alternating, 139]][q]
Out[12]=	-8 -6 2 -2 2 4 6 8 10 12 14 -1 + q - q + -- - q + 2 q - 3 q + 4 q - q + q + q - 2 q + 4 q 16 18 24 > 3 q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 139]][a, z]
Out[13]=	2 2 2 4 4 4 2 5 4 2 3 z 13 z 14 z 4 z 9 z 14 z 2 - -- + -- - -- + 5 z - ---- + ----- - ----- + 4 z - -- + ---- - ----- + 6 4 2 6 4 2 6 4 2 a a a a a a a a a 6 6 8 6 2 z 6 z z > z + ---- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 139]][a, z]
Out[14]=	2 2 2 2 5 4 3 z 5 z 3 z 2 z 2 2 z 4 z 19 z 2 + -- + -- + -- - --- - --- - --- - --- - a z - 4 z + ---- - ---- - ----- - 6 4 2 7 5 3 a 8 6 4 a a a a a a a a a 2 3 3 3 3 3 19 z 2 2 2 z 9 z 17 z 4 z 5 z 3 4 > ----- + 2 a z - ---- + ---- + ----- + ---- + ---- + 7 a z + 7 z - 2 9 7 5 3 a a a a a a 4 4 4 4 5 5 5 5 5 6 z 8 z 29 z 25 z 2 4 z 11 z 12 z 4 z 5 z > ---- + ---- + ----- + ----- - 3 a z + -- - ----- - ----- + ---- - ---- - 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 7 7 7 5 6 3 z 9 z 21 z 20 z 2 6 5 z z 10 z > 9 a z - 10 z + ---- - ---- - ----- - ----- + a z + ---- + -- - ----- - 8 6 4 2 7 5 3 a a a a a a a 7 8 8 8 9 9 9 10 10 3 z 7 8 5 z 7 z 6 z 3 z 6 z 3 z z z > ---- + 3 a z + 4 z + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- a 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 139]], Vassiliev[3][Knot[11, Alternating, 139]]}
Out[15]=	{1, 3}
In[16]:=	Kh[Knot[11, Alternating, 139]][q, t]
Out[16]=	3 1 2 1 4 2 6 4 q 3 8 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 8 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 > 7 q t + 8 q t + 8 q t + 6 q t + 8 q t + 4 q t + 6 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a139

K11a138 K11a140