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

K11a110 K11a112

Knotscape

This page is passe. Go here instead!    The Knot K11a111 Visit K11a111's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 16 2 22 20 18 8 6 12

Alexander Polynomial:

2t-3 - 10t-2 + 24t-1 - 31 + 24t - 10t2 + 2t3

Conway Polynomial:

1 + 2z2 + 2z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{10117, K11a23, ...}

Determinant and Signature:

{103, 2}

Jones Polynomial:

- q-4 + 3q-3 - 6q-2 + 10q-1 - 13 + 16q - 16q2 + 15q3 - 11q4 + 7q5 - 4q6 + q7

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

{K11a68, ...}

A2 (sl(3)) Invariant:

- q-12 + q-10 - q-8 - q-6 + 3q-4 - 2q-2 + 2 + 2q2 - q4 + 3q6 - 2q8 + 2q10 - 2q14 + 2q16 - 2q18 - q20 + q22

HOMFLY-PT Polynomial:

a-6z2 - a-4 - 3a-4z2 - 2a-4z4 + a-2 + 2a-2z2 + 2a-2z4 + a-2z6 + 2 + 4z2 + 3z4 + z6 - a2 - 2a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - 3a-7z3 + 4a-7z5 + 2a-6z2 - 7a-6z4 + 7a-6z6 + a-5z - a-5z3 - 7a-5z5 + 8a-5z7 - a-4 + 5a-4z2 - 6a-4z4 - 5a-4z6 + 7a-4z8 - a-3z + 12a-3z3 - 19a-3z5 + 3a-3z7 + 4a-3z9 - a-2 - 4a-2z2 + 24a-2z4 - 33a-2z6 + 11a-2z8 + a-2z10 - 5a-1z + 17a-1z3 - 7a-1z5 - 13a-1z7 + 7a-1z9 + 2 - 14z2 + 37z4 - 33z6 + 7z8 + z10 - 5az + 12az3 - 3az5 - 7az7 + 3az9 + a2 - 7a2z2 + 15a2z4 - 12a2z6 + 3a2z8 - 2a3z + 5a3z3 - 4a3z5 + a3z7

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=2 is the signature of 11111. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     3   j = 11                   4 1   j = 9                 7 3     j = 7               8 4       j = 5             8 7         j = 3           8 8           j = 1         6 9             j = -1       4 7               j = -3     2 6                 j = -5   1 4                   j = -7   2                     j = -9 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, 111]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 111]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 5, 15, 6], X[16, 8, 17, 7], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[20, 13, 21, 14], X[18, 16, 19, 15], > X[8, 18, 9, 17], X[6, 19, 7, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 111]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -9, 5, -2, 6, -11, 7, -3, 8, -4, 9, -8, 10, > -7, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 111]]
Out[5]=	DTCode[4, 10, 14, 16, 2, 22, 20, 18, 8, 6, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 111]][t]
Out[6]=	2 10 24 2 3 -31 + -- - -- + -- + 24 t - 10 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 111]][z]
Out[7]=	2 4 6 1 + 2 z + 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 117], Knot[11, Alternating, 23], Knot[11, Alternating, 111]}
In[9]:=	{KnotDet[Knot[11, Alternating, 111]], KnotSignature[Knot[11, Alternating, 111]]}
Out[9]=	{103, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 111]][q]
Out[10]=	-4 3 6 10 2 3 4 5 6 7 -13 - q + -- - -- + -- + 16 q - 16 q + 15 q - 11 q + 7 q - 4 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 68], Knot[11, Alternating, 111]}
In[12]:=	A2Invariant[Knot[11, Alternating, 111]][q]
Out[12]=	-12 -10 -8 -6 3 2 2 4 6 8 10 2 - q + q - q - q + -- - -- + 2 q - q + 3 q - 2 q + 2 q - 4 2 q q 14 16 18 20 22 > 2 q + 2 q - 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 111]][a, z]
Out[13]=	2 2 2 4 4 -4 -2 2 2 z 3 z 2 z 2 2 4 2 z 2 z 2 - a + a - a + 4 z + -- - ---- + ---- - 2 a z + 3 z - ---- + ---- - 6 4 2 4 2 a a a a a 6 2 4 6 z > a z + z + -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 111]][a, z]
Out[14]=	2 2 -4 -2 2 z z 5 z 3 2 2 z 5 z 2 - a - a + a + -- - -- - --- - 5 a z - 2 a z - 14 z + ---- + ---- - 5 3 a 6 4 a a a a 2 3 3 3 3 4 z 2 2 3 z z 12 z 17 z 3 3 3 4 > ---- - 7 a z - ---- - -- + ----- + ----- + 12 a z + 5 a z + 37 z + 2 7 5 3 a a a a a 4 4 4 4 5 5 5 5 z 7 z 6 z 24 z 2 4 4 z 7 z 19 z 7 z 5 > -- - ---- - ---- + ----- + 15 a z + ---- - ---- - ----- - ---- - 3 a z - 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 7 7 7 3 5 6 7 z 5 z 33 z 2 6 8 z 3 z 13 z > 4 a z - 33 z + ---- - ---- - ----- - 12 a z + ---- + ---- - ----- - 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 7 z 11 z 2 8 4 z 7 z 9 > 7 a z + a z + 7 z + ---- + ----- + 3 a z + ---- + ---- + 3 a z + 4 2 3 a a a a 10 10 z > z + --- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 111]], Vassiliev[3][Knot[11, Alternating, 111]]}
Out[15]=	{2, 1}
In[16]:=	Kh[Knot[11, Alternating, 111]][q, t]
Out[16]=	3 1 2 1 4 2 6 4 7 6 q 9 q + 8 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 > 8 q t + 8 q t + 7 q t + 8 q t + 4 q t + 7 q t + 3 q t + 11 4 11 5 13 5 15 6 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a111

K11a110 K11a112