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10150 10152

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KnotPlot

PD Presentation:

X1425 X3849 X12,6,13,5 X9,17,10,16 X17,1,18,20 X13,19,14,18 X19,15,20,14 X15,11,16,10 X6,12,7,11 X7283

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 -12 2 16 -6 18 10 20 14

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 Chiral 2 3 3 / NotAvailable 1

Alexander Polynomial:

t-3 - 4t-2 + 10t-1 - 13 + 10t - 4t2 + t3

Conway Polynomial:

1 + 3z2 + 2z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{K11n54, K11n129, ...}

Determinant and Signature:

{43, 2}

Jones Polynomial:

- q-2 + 3q-1 - 5 + 7q - 7q2 + 8q3 - 6q4 + 4q5 - 2q6

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

{...}

A2 (sl(3)) Invariant:

- q-6 + q-4 - q-2 + 2q2 - q4 + 3q6 + 2q10 + q12 - q14 + q16 - 2q18 - q20

HOMFLY-PT Polynomial:

- a-6 - a-4z2 - a-4z4 + 3a-2 + 6a-2z2 + 4a-2z4 + a-2z6 - 1 - 2z2 - z4

Kauffman Polynomial:

- 3a-7z + 3a-7z3 + a-6 - 2a-6z2 + 2a-6z4 + a-6z6 - 3a-5z + 5a-5z3 - 2a-5z5 + 2a-5z7 + 4a-4z2 - 6a-4z4 + 3a-4z6 + a-4z8 + a-3z + a-3z3 - 7a-3z5 + 5a-3z7 - 3a-2 + 10a-2z2 - 15a-2z4 + 5a-2z6 + a-2z8 + 2a-1z - 3a-1z3 - 4a-1z5 + 3a-1z7 - 1 + 4z2 - 7z4 + 3z6 + az - 2az3 + az5

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

{3, 4}

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 10151. 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 j = 13                 2 j = 11               2   j = 9             4 2   j = 7           4 2     j = 5         3 4       j = 3       4 4         j = 1     2 4           j = -1   1 3             j = -3   2               j = -5 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-7 - 3q-6 + 10q-4 - 12q-3 - 8q-2 + 30q-1 - 17 - 27q + 49q2 - 12q3 - 46q4 + 57q5 - 2q6 - 55q7 + 51q8 + 6q9 - 47q10 + 31q11 + 10q12 - 26q13 + 10q14 + 6q15 - 7q16 + q17 + q18
3	- q-15 + 3q-14 - 5q-12 - 5q-11 + 12q-10 + 15q-9 - 20q-8 - 32q-7 + 20q-6 + 60q-5 - 10q-4 - 92q-3 - 13q-2 + 120q-1 + 49 - 137q - 99q2 + 150q3 + 142q4 - 141q5 - 191q6 + 135q7 + 223q8 - 110q9 - 258q10 + 98q11 + 265q12 - 65q13 - 277q14 + 46q15 + 257q16 - 9q17 - 238q18 - 13q19 + 195q20 + 36q21 - 147q22 - 49q23 + 99q24 + 47q25 - 56q26 - 38q27 + 27q28 + 22q29 - 6q30 - 15q31 + 5q32 + 2q33 + 2q34 - 2q35
4	q-26 - 3q-25 + 5q-23 + 5q-21 - 19q-20 - 8q-19 + 20q-18 + 15q-17 + 38q-16 - 61q-15 - 64q-14 + 11q-13 + 52q-12 + 171q-11 - 60q-10 - 175q-9 - 130q-8 + 409q-6 + 121q-5 - 191q-4 - 394q-3 - 294q-2 + 558q-1 + 461 + 62q - 583q2 - 784q3 + 449q4 + 756q5 + 519q6 - 553q7 - 1245q8 + 140q9 + 873q10 + 978q11 - 368q12 - 1540q13 - 195q14 + 848q15 + 1305q16 - 148q17 - 1657q18 - 473q19 + 735q20 + 1475q21 + 79q22 - 1592q23 - 685q24 + 514q25 + 1457q26 + 328q27 - 1295q28 - 795q29 + 179q30 + 1196q31 + 519q32 - 794q33 - 700q34 - 142q35 + 725q36 + 515q37 - 290q38 - 414q39 - 260q40 + 271q41 + 315q42 - 18q43 - 131q44 - 168q45 + 37q46 + 107q47 + 27q48 - 9q49 - 51q50 - 7q51 + 17q52 + 7q53 + 4q54 - 6q55 - 3q56 + q57 + q58
5	- q-40 + 3q-39 - 5q-37 + 2q-34 + 12q-33 + 8q-32 - 20q-31 - 24q-30 - 12q-29 + 12q-28 + 56q-27 + 62q-26 - 6q-25 - 106q-24 - 132q-23 - 54q-22 + 116q-21 + 268q-20 + 212q-19 - 76q-18 - 406q-17 - 464q-16 - 126q-15 + 458q-14 + 819q-13 + 528q-12 - 340q-11 - 1148q-10 - 1114q-9 - 79q-8 + 1306q-7 + 1834q-6 + 806q-5 - 1166q-4 - 2498q-3 - 1809q-2 + 649q-1 + 2942 + 2944q + 267q2 - 3078q3 - 4063q4 - 1410q5 + 2820q6 + 4966q7 + 2758q8 - 2276q9 - 5657q10 - 4009q11 + 1502q12 + 6016q13 + 5202q14 - 640q15 - 6228q16 - 6116q17 - 209q18 + 6187q19 + 6930q20 + 971q21 - 6134q22 - 7456q23 - 1656q24 + 5910q25 + 7951q26 + 2252q27 - 5723q28 - 8185q29 - 2827q30 + 5307q31 + 8418q32 + 3383q33 - 4864q34 - 8344q35 - 3939q36 + 4098q37 + 8157q38 + 4461q39 - 3238q40 - 7587q41 - 4843q42 + 2119q43 + 6757q44 + 5020q45 - 990q46 - 5597q47 - 4887q48 - 67q49 + 4240q50 + 4421q51 + 886q52 - 2850q53 - 3665q54 - 1360q55 + 1616q56 + 2730q57 + 1477q58 - 678q59 - 1796q60 - 1290q61 + 70q62 + 1030q63 + 958q64 + 175q65 - 473q66 - 572q67 - 262q68 + 168q69 + 325q70 + 158q71 - 24q72 - 113q73 - 115q74 - 18q75 + 60q76 + 34q77 + 10q78 - q79 - 22q80 - 7q81 + 5q82 + 4q83 + 2q85 - 2q86
6	q-57 - 3q-56 + 5q-54 - 7q-51 + 5q-50 - 12q-49 - 8q-48 + 29q-47 + 15q-46 + 12q-45 - 29q-44 - 7q-43 - 65q-42 - 56q-41 + 72q-40 + 102q-39 + 131q-38 + 3q-37 + q-36 - 274q-35 - 351q-34 - 73q-33 + 183q-32 + 526q-31 + 457q-30 + 490q-29 - 375q-28 - 1070q-27 - 1072q-26 - 589q-25 + 571q-24 + 1415q-23 + 2402q-22 + 1066q-21 - 952q-20 - 2721q-19 - 3355q-18 - 1901q-17 + 740q-16 + 4911q-15 + 5254q-14 + 2900q-13 - 1807q-12 - 6366q-11 - 7871q-10 - 5026q-9 + 3615q-8 + 9321q-7 + 10909q-6 + 5443q-5 - 4234q-4 - 13277q-3 - 15429q-2 - 5189q-1 + 7230 + 17892q + 17586q2 + 6209q3 - 11863q4 - 24437q5 - 19019q6 - 3343q7 + 17824q8 + 28161q9 + 21452q10 - 2302q11 - 26575q12 - 31284q13 - 18165q14 + 10341q15 + 32471q16 + 35021q17 + 10784q18 - 22209q19 - 37992q20 - 31223q21 + 47q22 + 31297q23 + 43550q24 + 22101q25 - 15505q26 - 39980q27 - 39836q28 - 8734q29 + 28065q30 + 47813q31 + 29765q32 - 9656q33 - 39861q34 - 44855q35 - 14993q36 + 24848q37 + 49793q38 + 34917q39 - 4769q40 - 38806q41 - 47977q42 - 20269q43 + 20995q44 + 50048q45 + 39177q46 + 1077q47 - 35575q48 - 49357q49 - 26107q50 + 14301q51 + 46837q52 + 42256q53 + 9250q54 - 27737q55 - 46734q56 - 31588q57 + 3743q58 + 37568q59 + 41183q60 + 17946q61 - 14781q62 - 37349q63 - 32841q64 - 7769q65 + 22463q66 + 32959q67 + 22214q68 - 864q69 - 22143q70 - 26637q71 - 14315q72 + 6932q73 + 19187q74 + 18698q75 + 7382q76 - 7357q77 - 15176q78 - 12818q79 - 2214q80 + 6481q81 + 10229q82 + 7577q83 + 735q84 - 5086q85 - 6799q86 - 3635q87 + 94q88 + 3118q89 + 3803q90 + 2055q91 - 429q92 - 2000q93 - 1710q94 - 928q95 + 231q96 + 970q97 + 915q98 + 327q99 - 243q100 - 341q101 - 357q102 - 146q103 + 92q104 + 183q105 + 113q106 + 6q107 - 17q108 - 48q109 - 41q110 - 8q111 + 21q112 + 13q113 + 2q114 + q115 - 2q116 - 2q117 - 3q118 + q119 + q120

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, 151]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[12, 6, 13, 5], X[9, 17, 10, 16], > X[17, 1, 18, 20], X[13, 19, 14, 18], X[19, 15, 20, 14], X[15, 11, 16, 10], > X[6, 12, 7, 11], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 151]]
Out[3]=	GaussCode[-1, 10, -2, 1, 3, -9, -10, 2, -4, 8, 9, -3, -6, 7, -8, 4, -5, 6, -7, > 5]
In[4]:=	DTCode[Knot[10, 151]]
Out[4]=	DTCode[4, 8, -12, 2, 16, -6, 18, 10, 20, 14]
In[5]:=	br = BR[Knot[10, 151]]
Out[5]=	BR[4, {1, 1, 1, 2, -1, -1, 3, -2, 1, 3, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 151]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 151]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 151]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 151]][t]
Out[10]=	-3 4 10 2 3 -13 + t - -- + -- + 10 t - 4 t + t 2 t t
In[11]:=	Conway[Knot[10, 151]][z]
Out[11]=	2 4 6 1 + 3 z + 2 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 151], Knot[11, NonAlternating, 54], Knot[11, NonAlternating, 129]}
In[13]:=	{KnotDet[Knot[10, 151]], KnotSignature[Knot[10, 151]]}
Out[13]=	{43, 2}
In[14]:=	Jones[Knot[10, 151]][q]
Out[14]=	-2 3 2 3 4 5 6 -5 - q + - + 7 q - 7 q + 8 q - 6 q + 4 q - 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 151]}
In[16]:=	A2Invariant[Knot[10, 151]][q]
Out[16]=	-6 -4 -2 2 4 6 10 12 14 16 18 20 -q + q - q + 2 q - q + 3 q + 2 q + q - q + q - 2 q - q
In[17]:=	HOMFLYPT[Knot[10, 151]][a, z]
Out[17]=	2 2 4 4 6 -6 3 2 z 6 z 4 z 4 z z -1 - a + -- - 2 z - -- + ---- - z - -- + ---- + -- 2 4 2 4 2 2 a a a a a a
In[18]:=	Kauffman[Knot[10, 151]][a, z]
Out[18]=	2 2 2 -6 3 3 z 3 z z 2 z 2 2 z 4 z 10 z -1 + a - -- - --- - --- + -- + --- + a z + 4 z - ---- + ---- + ----- + 2 7 5 3 a 6 4 2 a a a a a a a 3 3 3 3 4 4 4 5 3 z 5 z z 3 z 3 4 2 z 6 z 15 z 2 z > ---- + ---- + -- - ---- - 2 a z - 7 z + ---- - ---- - ----- - ---- - 7 5 3 a 6 4 2 5 a a a a a a a 5 5 6 6 6 7 7 7 8 8 7 z 4 z 5 6 z 3 z 5 z 2 z 5 z 3 z z z > ---- - ---- + a z + 3 z + -- + ---- + ---- + ---- + ---- + ---- + -- + -- 3 a 6 4 2 5 3 a 4 2 a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 151]], Vassiliev[3][Knot[10, 151]]}
Out[19]=	{3, 4}
In[20]:=	Kh[Knot[10, 151]][q, t]
Out[20]=	3 1 2 1 3 2 q 3 5 5 2 4 q + 4 q + ----- + ----- + ---- + --- + --- + 4 q t + 3 q t + 4 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 13 5 > 4 q t + 2 q t + 4 q t + 2 q t + 2 q t + 2 q t
In[21]:=	ColouredJones[Knot[10, 151], 2][q]
Out[21]=	-7 3 10 12 8 30 2 3 4 5 -17 + q - -- + -- - -- - -- + -- - 27 q + 49 q - 12 q - 46 q + 57 q - 6 4 3 2 q q q q q 6 7 8 9 10 11 12 13 14 > 2 q - 55 q + 51 q + 6 q - 47 q + 31 q + 10 q - 26 q + 10 q + 15 16 17 18 > 6 q - 7 q + q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10151

10150 10152