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10121 10123

KnotPlot

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KnotPlot

PD Presentation:

X1627 X7,15,8,14 X15,2,16,3 X5,12,6,13 X9,19,10,18 X3,11,4,10 X17,5,18,4 X19,9,20,8 X11,16,12,17 X13,1,14,20

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 12 14 18 16 20 2 4 8

Minimum Braid Representative: Length is 11, width is 4 Braid index is 4

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index Reversible 2 3 3 / NotAvailable 2

Alexander Polynomial:

- 2t-3 + 11t-2 - 24t-1 + 31 - 24t + 11t2 - 2t3

Conway Polynomial:

1 + 2z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11n185, ...}

Determinant and Signature:

{105, 0}

Jones Polynomial:

q-4 - 4q-3 + 8q-2 - 13q-1 + 17 - 17q + 17q2 - 13q3 + 9q4 - 5q5 + q6

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

{...}

A2 (sl(3)) Invariant:

q-12 - 2q-10 + q-8 + q-6 - 4q-4 + 3q-2 - 2 + 2q2 + 3q4 + 5q8 - 3q10 - 3q16 + q18

HOMFLY-PT Polynomial:

- 2a-4 + a-4z4 + 4a-2 + 3a-2z2 - a-2z4 - a-2z6 - 1 - 2z2 - 2z4 - z6 + a2z2 + a2z4

Kauffman Polynomial:

- a-6z4 + a-6z6 + 2a-5z + 4a-5z3 - 11a-5z5 + 5a-5z7 - 2a-4 + 12a-4z4 - 20a-4z6 + 8a-4z8 + 2a-3z + 14a-3z3 - 25a-3z5 + 3a-3z7 + 4a-3z9 - 4a-2 + 24a-2z4 - 42a-2z6 + 18a-2z8 + 18a-1z3 - 32a-1z5 + 9a-1z7 + 4a-1z9 - 1 + 2z2 + 3z4 - 13z6 + 10z8 + 6az3 - 14az5 + 11az7 + 2a2z2 - 7a2z4 + 8a2z6 - 2a3z3 + 4a3z5 + a4z4

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

{2, 2}

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 10122. 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 j = 13                     1 j = 11                   4   j = 9                 5 1   j = 7               8 4     j = 5             9 5       j = 3           8 8         j = 1         9 9           j = -1       5 9             j = -3     3 8               j = -5   1 5                 j = -7   3                   j = -9 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-12 - 4q-11 + 4q-10 + 7q-9 - 25q-8 + 25q-7 + 23q-6 - 88q-5 + 66q-4 + 74q-3 - 183q-2 + 85q-1 + 152 - 246q + 59q2 + 207q3 - 239q4 + 7q5 + 212q6 - 174q7 - 41q8 + 164q9 - 83q10 - 57q11 + 84q12 - 15q13 - 33q14 + 20q15 + 3q16 - 5q17 + q18
3	q-24 - 4q-23 + 4q-22 + 3q-21 - 5q-20 - 7q-19 + 15q-18 + 10q-17 - 50q-16 + q-15 + 111q-14 + 6q-13 - 243q-12 - 39q-11 + 438q-10 + 152q-9 - 683q-8 - 360q-7 + 910q-6 + 677q-5 - 1080q-4 - 1050q-3 + 1145q-2 + 1404q-1 - 1066 - 1730q + 929q2 + 1920q3 - 682q4 - 2048q5 + 443q6 + 2034q7 - 139q8 - 1978q9 - 125q10 + 1801q11 + 413q12 - 1578q13 - 631q14 + 1254q15 + 799q16 - 897q17 - 856q18 + 534q19 + 788q20 - 208q21 - 638q22 - 9q23 + 429q24 + 121q25 - 241q26 - 125q27 + 92q28 + 97q29 - 29q30 - 42q31 + 14q33 + 3q34 - 5q35 + q36
4	q-40 - 4q-39 + 4q-38 + 3q-37 - 9q-36 + 13q-35 - 17q-34 + 12q-33 - 2q-32 - 43q-31 + 97q-30 - 13q-29 - 5q-28 - 141q-27 - 215q-26 + 451q-25 + 342q-24 + 41q-23 - 882q-22 - 1214q-21 + 1092q-20 + 1994q-19 + 1225q-18 - 2335q-17 - 4606q-16 + 477q-15 + 5148q-14 + 5657q-13 - 2413q-12 - 10460q-11 - 3926q-10 + 7028q-9 + 13052q-8 + 1703q-7 - 15187q-6 - 11417q-5 + 4533q-4 + 19206q-3 + 8955q-2 - 15351q-1 - 17650 - 1299q + 20794q2 + 15253q3 - 11705q4 - 19850q5 - 7057q6 + 18492q7 + 18435q8 - 6782q9 - 18735q10 - 11277q11 + 14190q12 + 19134q13 - 1524q14 - 15584q15 - 14277q16 + 8401q17 + 17807q18 + 4000q19 - 10334q20 - 15524q21 + 1429q22 + 13620q23 + 8247q24 - 3251q25 - 13248q26 - 4473q27 + 6684q28 + 8585q29 + 2969q30 - 7414q31 - 6159q32 + 171q33 + 4858q34 + 4943q35 - 1547q36 - 3644q37 - 2369q38 + 747q39 + 2997q40 + 880q41 - 673q42 - 1468q43 - 701q44 + 741q45 + 586q46 + 281q47 - 285q48 - 370q49 + 14q50 + 73q51 + 126q52 + 8q53 - 56q54 - 9q55 - 6q56 + 14q57 + 3q58 - 5q59 + q60

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]:=	PD[Knot[10, 122]]
Out[2]=	PD[X[1, 6, 2, 7], X[7, 15, 8, 14], X[15, 2, 16, 3], X[5, 12, 6, 13], > X[9, 19, 10, 18], X[3, 11, 4, 10], X[17, 5, 18, 4], X[19, 9, 20, 8], > X[11, 16, 12, 17], X[13, 1, 14, 20]]
In[3]:=	GaussCode[Knot[10, 122]]
Out[3]=	GaussCode[-1, 3, -6, 7, -4, 1, -2, 8, -5, 6, -9, 4, -10, 2, -3, 9, -7, 5, -8, > 10]
In[4]:=	DTCode[Knot[10, 122]]
Out[4]=	DTCode[6, 10, 12, 14, 18, 16, 20, 2, 4, 8]
In[5]:=	br = BR[Knot[10, 122]]
Out[5]=	BR[4, {1, 1, 2, -3, 2, -1, -3, 2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 122]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 122]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 122]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 122]][t]
Out[10]=	2 11 24 2 3 31 - -- + -- - -- - 24 t + 11 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 122]][z]
Out[11]=	2 4 6 1 + 2 z - z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 122], Knot[11, NonAlternating, 185]}
In[13]:=	{KnotDet[Knot[10, 122]], KnotSignature[Knot[10, 122]]}
Out[13]=	{105, 0}
In[14]:=	Jones[Knot[10, 122]][q]
Out[14]=	-4 4 8 13 2 3 4 5 6 17 + q - -- + -- - -- - 17 q + 17 q - 13 q + 9 q - 5 q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 122]}
In[16]:=	A2Invariant[Knot[10, 122]][q]
Out[16]=	-12 2 -8 -6 4 3 2 4 8 10 16 18 -2 + q - --- + q + q - -- + -- + 2 q + 3 q + 5 q - 3 q - 3 q + q 10 4 2 q q q
In[17]:=	HOMFLYPT[Knot[10, 122]][a, z]
Out[17]=	2 4 4 6 2 4 2 3 z 2 2 4 z z 2 4 6 z -1 - -- + -- - 2 z + ---- + a z - 2 z + -- - -- + a z - z - -- 4 2 2 4 2 2 a a a a a a
In[18]:=	Kauffman[Knot[10, 122]][a, z]
Out[18]=	3 3 3 2 4 2 z 2 z 2 2 2 4 z 14 z 18 z 3 -1 - -- - -- + --- + --- + 2 z + 2 a z + ---- + ----- + ----- + 6 a z - 4 2 5 3 5 3 a a a a a a a 4 4 4 5 5 3 3 4 z 12 z 24 z 2 4 4 4 11 z 25 z > 2 a z + 3 z - -- + ----- + ----- - 7 a z + a z - ----- - ----- - 6 4 2 5 3 a a a a a 5 6 6 6 7 32 z 5 3 5 6 z 20 z 42 z 2 6 5 z > ----- - 14 a z + 4 a z - 13 z + -- - ----- - ----- + 8 a z + ---- + a 6 4 2 5 a a a a 7 7 8 8 9 9 3 z 9 z 7 8 8 z 18 z 4 z 4 z > ---- + ---- + 11 a z + 10 z + ---- + ----- + ---- + ---- 3 a 4 2 3 a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 122]], Vassiliev[3][Knot[10, 122]]}
Out[19]=	{2, 2}
In[20]:=	Kh[Knot[10, 122]][q, t]
Out[20]=	9 1 3 1 5 3 8 5 3 - + 9 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 q t + 8 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 8 q t + 9 q t + 5 q t + 8 q t + 4 q t + 5 q t + q t + 11 5 13 6 > 4 q t + q t
In[21]:=	ColouredJones[Knot[10, 122], 2][q]
Out[21]=	-12 4 4 7 25 25 23 88 66 74 183 85 152 + q - --- + --- + -- - -- + -- + -- - -- + -- + -- - --- + -- - 246 q + 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 9 > 59 q + 207 q - 239 q + 7 q + 212 q - 174 q - 41 q + 164 q - 10 11 12 13 14 15 16 17 18 > 83 q - 57 q + 84 q - 15 q - 33 q + 20 q + 3 q - 5 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10122

10121 10123