9 49

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9 48.gif

9_48

10 1.gif

10_1

9 49.gif Visit 9 49's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 49's page at Knotilus!

Visit 9 49's page at the original Knot Atlas!

9 49 Quick Notes


9 49 Further Notes and Views

Knot presentations

Planar diagram presentation X6271 X12,8,13,7 X5,15,6,14 X3,11,4,10 X11,3,12,2 X15,5,16,4 X17,9,18,8 X9,17,10,16 X18,14,1,13
Gauss code 1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9
Dowker-Thistlethwaite code 6 -10 -14 12 -16 -2 18 -4 -8
Conway Notation [-20:-20:-20]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 2
Bridge index 3
Super bridge index [math]\displaystyle{ \{4,5\} }[/math]
Nakanishi index 2
Maximal Thurston-Bennequin number [3][-12]
Hyperbolic Volume 9.42707
A-Polynomial See Data:9 49/A-polynomial

[edit Notes for 9 49's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 2 }[/math]
Topological 4 genus [math]\displaystyle{ 2 }[/math]
Concordance genus [math]\displaystyle{ 2 }[/math]
Rasmussen s-Invariant -4

[edit Notes for 9 49's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 3 t^2-6 t+7-6 t^{-1} +3 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ 3 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{5,t+1\} }[/math]
Determinant and Signature { 25, 4 }
Jones polynomial [math]\displaystyle{ -2 q^9+3 q^8-4 q^7+5 q^6-4 q^5+4 q^4-2 q^3+q^2 }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^4 a^{-4} +2 z^4 a^{-6} +2 z^2 a^{-4} +6 z^2 a^{-6} -2 z^2 a^{-8} +4 a^{-6} -3 a^{-8} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^7 a^{-7} +z^7 a^{-9} +3 z^6 a^{-6} +4 z^6 a^{-8} +z^6 a^{-10} +2 z^5 a^{-5} +z^5 a^{-7} -z^5 a^{-9} +z^4 a^{-4} -8 z^4 a^{-6} -9 z^4 a^{-8} -3 z^3 a^{-5} -3 z^3 a^{-7} +3 z^3 a^{-9} +3 z^3 a^{-11} -2 z^2 a^{-4} +9 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +2 z a^{-7} -2 z a^{-9} -4 z a^{-11} -4 a^{-6} -3 a^{-8} }[/math]
The A2 invariant [math]\displaystyle{ q^{-6} - q^{-8} + q^{-10} + q^{-14} +3 q^{-16} + q^{-18} +2 q^{-20} - q^{-22} - q^{-24} - q^{-26} -2 q^{-28} }[/math]
The G2 invariant [math]\displaystyle{ q^{-30} - q^{-32} +2 q^{-34} -3 q^{-36} +2 q^{-38} - q^{-40} -2 q^{-42} +8 q^{-44} -10 q^{-46} +12 q^{-48} -7 q^{-50} - q^{-52} +10 q^{-54} -16 q^{-56} +19 q^{-58} -11 q^{-60} + q^{-62} +10 q^{-64} -14 q^{-66} +13 q^{-68} -2 q^{-70} -6 q^{-72} +14 q^{-74} -12 q^{-76} +4 q^{-78} +9 q^{-80} -15 q^{-82} +21 q^{-84} -16 q^{-86} +9 q^{-88} +5 q^{-90} -13 q^{-92} +22 q^{-94} -22 q^{-96} +16 q^{-98} -4 q^{-100} -7 q^{-102} +13 q^{-104} -16 q^{-106} +9 q^{-108} - q^{-110} -10 q^{-112} +10 q^{-114} -11 q^{-116} -2 q^{-118} +10 q^{-120} -20 q^{-122} +16 q^{-124} -9 q^{-126} -4 q^{-128} +11 q^{-130} -16 q^{-132} +15 q^{-134} -7 q^{-136} + q^{-138} +3 q^{-140} -7 q^{-142} +7 q^{-144} -2 q^{-146} + q^{-148} + q^{-150} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_49.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{-3} - q^{-5} +2 q^{-7} + q^{-11} + q^{-13} - q^{-15} + q^{-17} -2 q^{-19} }[/math]
2 [math]\displaystyle{ q^{-6} - q^{-8} +5 q^{-12} - q^{-14} -5 q^{-16} +5 q^{-18} +2 q^{-20} -5 q^{-22} +4 q^{-24} +4 q^{-26} -3 q^{-28} - q^{-30} + q^{-32} + q^{-34} -5 q^{-36} + q^{-38} +5 q^{-40} -6 q^{-42} -2 q^{-44} +5 q^{-46} -3 q^{-48} -3 q^{-50} +3 q^{-52} + q^{-54} }[/math]
3 [math]\displaystyle{ q^{-9} - q^{-11} +3 q^{-15} +3 q^{-17} -3 q^{-19} -7 q^{-21} +3 q^{-23} +14 q^{-25} +4 q^{-27} -14 q^{-29} -13 q^{-31} +14 q^{-33} +18 q^{-35} -9 q^{-37} -21 q^{-39} +4 q^{-41} +23 q^{-43} + q^{-45} -19 q^{-47} -3 q^{-49} +14 q^{-51} +6 q^{-53} -9 q^{-55} -9 q^{-57} +2 q^{-59} +8 q^{-61} + q^{-63} -14 q^{-65} -7 q^{-67} +14 q^{-69} +14 q^{-71} -16 q^{-73} -17 q^{-75} +12 q^{-77} +21 q^{-79} -6 q^{-81} -21 q^{-83} - q^{-85} +18 q^{-87} +5 q^{-89} -10 q^{-91} -7 q^{-93} +5 q^{-95} +8 q^{-97} - q^{-99} -2 q^{-101} -2 q^{-103} }[/math]
4 [math]\displaystyle{ q^{-12} - q^{-14} +3 q^{-18} + q^{-20} + q^{-22} -6 q^{-24} -4 q^{-26} +8 q^{-28} +10 q^{-30} +14 q^{-32} -12 q^{-34} -28 q^{-36} -13 q^{-38} +12 q^{-40} +51 q^{-42} +23 q^{-44} -29 q^{-46} -58 q^{-48} -37 q^{-50} +56 q^{-52} +76 q^{-54} +19 q^{-56} -66 q^{-58} -93 q^{-60} +10 q^{-62} +84 q^{-64} +69 q^{-66} -28 q^{-68} -97 q^{-70} -28 q^{-72} +51 q^{-74} +71 q^{-76} +4 q^{-78} -62 q^{-80} -38 q^{-82} +14 q^{-84} +45 q^{-86} +17 q^{-88} -25 q^{-90} -34 q^{-92} -9 q^{-94} +25 q^{-96} +32 q^{-98} +7 q^{-100} -43 q^{-102} -38 q^{-104} +13 q^{-106} +56 q^{-108} +43 q^{-110} -49 q^{-112} -73 q^{-114} -13 q^{-116} +71 q^{-118} +87 q^{-120} -25 q^{-122} -88 q^{-124} -57 q^{-126} +39 q^{-128} +99 q^{-130} +24 q^{-132} -50 q^{-134} -75 q^{-136} -17 q^{-138} +58 q^{-140} +45 q^{-142} +7 q^{-144} -37 q^{-146} -35 q^{-148} +5 q^{-150} +19 q^{-152} +20 q^{-154} - q^{-156} -13 q^{-158} -7 q^{-160} -3 q^{-162} +3 q^{-164} +3 q^{-166} + q^{-168} }[/math]
5 [math]\displaystyle{ q^{-15} - q^{-17} +3 q^{-21} + q^{-23} - q^{-25} -2 q^{-27} -3 q^{-29} +9 q^{-33} +14 q^{-35} +3 q^{-37} -10 q^{-39} -24 q^{-41} -23 q^{-43} +40 q^{-47} +59 q^{-49} +34 q^{-51} -28 q^{-53} -92 q^{-55} -97 q^{-57} -27 q^{-59} +96 q^{-61} +171 q^{-63} +117 q^{-65} -44 q^{-67} -206 q^{-69} -231 q^{-71} -64 q^{-73} +197 q^{-75} +325 q^{-77} +194 q^{-79} -114 q^{-81} -363 q^{-83} -326 q^{-85} -8 q^{-87} +339 q^{-89} +402 q^{-91} +127 q^{-93} -257 q^{-95} -424 q^{-97} -219 q^{-99} +159 q^{-101} +387 q^{-103} +263 q^{-105} -72 q^{-107} -310 q^{-109} -263 q^{-111} +5 q^{-113} +228 q^{-115} +229 q^{-117} +33 q^{-119} -156 q^{-121} -183 q^{-123} -56 q^{-125} +92 q^{-127} +141 q^{-129} +66 q^{-131} -52 q^{-133} -117 q^{-135} -81 q^{-137} +20 q^{-139} +108 q^{-141} +112 q^{-143} +16 q^{-145} -116 q^{-147} -156 q^{-149} -50 q^{-151} +121 q^{-153} +216 q^{-155} +116 q^{-157} -128 q^{-159} -281 q^{-161} -184 q^{-163} +100 q^{-165} +334 q^{-167} +279 q^{-169} -42 q^{-171} -353 q^{-173} -359 q^{-175} -46 q^{-177} +317 q^{-179} +417 q^{-181} +151 q^{-183} -229 q^{-185} -414 q^{-187} -254 q^{-189} +106 q^{-191} +349 q^{-193} +299 q^{-195} +25 q^{-197} -232 q^{-199} -292 q^{-201} -124 q^{-203} +107 q^{-205} +219 q^{-207} +160 q^{-209} +3 q^{-211} -120 q^{-213} -141 q^{-215} -61 q^{-217} +42 q^{-219} +82 q^{-221} +63 q^{-223} +12 q^{-225} -31 q^{-227} -46 q^{-229} -21 q^{-231} +3 q^{-233} +12 q^{-235} +17 q^{-237} +8 q^{-239} - q^{-241} -4 q^{-243} -2 q^{-245} -2 q^{-247} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-6} - q^{-8} + q^{-10} + q^{-14} +3 q^{-16} + q^{-18} +2 q^{-20} - q^{-22} - q^{-24} - q^{-26} -2 q^{-28} }[/math]
1,1 [math]\displaystyle{ q^{-12} -2 q^{-14} +4 q^{-16} -8 q^{-18} +19 q^{-20} -22 q^{-22} +36 q^{-24} -42 q^{-26} +48 q^{-28} -40 q^{-30} +34 q^{-32} -10 q^{-34} -6 q^{-36} +38 q^{-38} -56 q^{-40} +70 q^{-42} -87 q^{-44} +78 q^{-46} -84 q^{-48} +58 q^{-50} -45 q^{-52} +18 q^{-54} +10 q^{-56} -22 q^{-58} +45 q^{-60} -48 q^{-62} +48 q^{-64} -42 q^{-66} +25 q^{-68} -22 q^{-70} +10 q^{-72} -2 q^{-74} +2 q^{-76} +2 q^{-78} }[/math]
2,0 [math]\displaystyle{ q^{-12} - q^{-14} - q^{-16} +3 q^{-18} +3 q^{-20} - q^{-22} - q^{-24} +2 q^{-26} +2 q^{-28} -2 q^{-30} + q^{-32} +4 q^{-34} +2 q^{-36} +3 q^{-38} +5 q^{-40} +2 q^{-42} - q^{-44} - q^{-46} -2 q^{-48} -6 q^{-50} -4 q^{-52} - q^{-54} - q^{-56} -5 q^{-58} - q^{-60} + q^{-62} +3 q^{-70} + q^{-72} + q^{-74} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{-12} - q^{-14} +2 q^{-18} -3 q^{-20} +2 q^{-22} +6 q^{-24} -2 q^{-26} +5 q^{-28} +6 q^{-30} + q^{-34} +2 q^{-36} - q^{-38} -2 q^{-40} -3 q^{-42} -3 q^{-46} -6 q^{-48} +2 q^{-50} -2 q^{-52} -5 q^{-54} +4 q^{-56} + q^{-58} - q^{-60} +3 q^{-62} }[/math]
1,0,0 [math]\displaystyle{ q^{-9} - q^{-11} + q^{-13} - q^{-15} + q^{-17} + q^{-19} +3 q^{-21} +3 q^{-23} +2 q^{-25} +2 q^{-27} - q^{-29} - q^{-31} -3 q^{-33} - q^{-35} -2 q^{-37} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{-18} - q^{-20} + q^{-24} - q^{-26} - q^{-28} +4 q^{-30} +2 q^{-32} - q^{-34} +4 q^{-36} +7 q^{-38} +3 q^{-40} +2 q^{-42} +8 q^{-44} +8 q^{-46} +3 q^{-50} +5 q^{-52} -3 q^{-54} -6 q^{-56} -2 q^{-58} -8 q^{-60} -13 q^{-62} -8 q^{-64} -5 q^{-66} -6 q^{-68} -3 q^{-70} +6 q^{-72} +6 q^{-74} +2 q^{-76} +3 q^{-78} +4 q^{-80} - q^{-84} }[/math]
1,0,0,0 [math]\displaystyle{ q^{-12} - q^{-14} + q^{-16} - q^{-18} + q^{-22} + q^{-24} +3 q^{-26} +3 q^{-28} +4 q^{-30} +2 q^{-32} +2 q^{-34} - q^{-36} - q^{-38} -3 q^{-40} -3 q^{-42} - q^{-44} -2 q^{-46} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{-12} - q^{-14} +2 q^{-16} -4 q^{-18} +5 q^{-20} -4 q^{-22} +4 q^{-24} +3 q^{-28} +2 q^{-30} -2 q^{-32} +7 q^{-34} -8 q^{-36} +7 q^{-38} -8 q^{-40} +5 q^{-42} -6 q^{-44} +3 q^{-46} -2 q^{-50} +4 q^{-52} -3 q^{-54} +4 q^{-56} -5 q^{-58} +3 q^{-60} -3 q^{-62} }[/math]
1,0 [math]\displaystyle{ q^{-18} - q^{-22} - q^{-24} + q^{-26} +3 q^{-28} -4 q^{-32} - q^{-34} +5 q^{-36} +5 q^{-38} -2 q^{-40} -2 q^{-42} + q^{-44} +6 q^{-46} +2 q^{-48} -2 q^{-50} - q^{-52} +4 q^{-54} +2 q^{-56} -2 q^{-58} -2 q^{-60} + q^{-62} +3 q^{-64} - q^{-66} -4 q^{-68} -2 q^{-70} +2 q^{-72} -5 q^{-76} -4 q^{-78} +2 q^{-80} +3 q^{-82} -3 q^{-84} -5 q^{-86} - q^{-88} +4 q^{-90} +3 q^{-92} -2 q^{-94} -2 q^{-96} + q^{-98} +3 q^{-100} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{-18} - q^{-20} + q^{-22} -2 q^{-24} +3 q^{-26} -4 q^{-28} +4 q^{-30} -2 q^{-32} +5 q^{-34} + q^{-36} +4 q^{-38} +4 q^{-40} +5 q^{-42} +6 q^{-44} -2 q^{-46} +6 q^{-48} -5 q^{-50} +5 q^{-52} -8 q^{-54} +2 q^{-56} -8 q^{-58} +3 q^{-60} -5 q^{-62} -4 q^{-66} -2 q^{-68} + q^{-70} -4 q^{-72} + q^{-74} -4 q^{-76} +4 q^{-78} -2 q^{-80} +4 q^{-82} -2 q^{-84} +3 q^{-86} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-30} - q^{-32} +2 q^{-34} -3 q^{-36} +2 q^{-38} - q^{-40} -2 q^{-42} +8 q^{-44} -10 q^{-46} +12 q^{-48} -7 q^{-50} - q^{-52} +10 q^{-54} -16 q^{-56} +19 q^{-58} -11 q^{-60} + q^{-62} +10 q^{-64} -14 q^{-66} +13 q^{-68} -2 q^{-70} -6 q^{-72} +14 q^{-74} -12 q^{-76} +4 q^{-78} +9 q^{-80} -15 q^{-82} +21 q^{-84} -16 q^{-86} +9 q^{-88} +5 q^{-90} -13 q^{-92} +22 q^{-94} -22 q^{-96} +16 q^{-98} -4 q^{-100} -7 q^{-102} +13 q^{-104} -16 q^{-106} +9 q^{-108} - q^{-110} -10 q^{-112} +10 q^{-114} -11 q^{-116} -2 q^{-118} +10 q^{-120} -20 q^{-122} +16 q^{-124} -9 q^{-126} -4 q^{-128} +11 q^{-130} -16 q^{-132} +15 q^{-134} -7 q^{-136} + q^{-138} +3 q^{-140} -7 q^{-142} +7 q^{-144} -2 q^{-146} + q^{-148} + q^{-150} }[/math]

.

Computer Talk
The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["9 49"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 3 t^2-6 t+7-6 t^{-1} +3 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 3 z^4+6 z^2+1 }[/math]
In[6]:= Alexander[K, 2][t]
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
Out[6]= [math]\displaystyle{ \{5,t+1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 25, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -2 q^9+3 q^8-4 q^7+5 q^6-4 q^5+4 q^4-2 q^3+q^2 }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^4 a^{-4} +2 z^4 a^{-6} +2 z^2 a^{-4} +6 z^2 a^{-6} -2 z^2 a^{-8} +4 a^{-6} -3 a^{-8} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^7 a^{-7} +z^7 a^{-9} +3 z^6 a^{-6} +4 z^6 a^{-8} +z^6 a^{-10} +2 z^5 a^{-5} +z^5 a^{-7} -z^5 a^{-9} +z^4 a^{-4} -8 z^4 a^{-6} -9 z^4 a^{-8} -3 z^3 a^{-5} -3 z^3 a^{-7} +3 z^3 a^{-9} +3 z^3 a^{-11} -2 z^2 a^{-4} +9 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +2 z a^{-7} -2 z a^{-9} -4 z a^{-11} -4 a^{-6} -3 a^{-8} }[/math]

Vassiliev invariants

V2 and V3: (6, 14)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
[math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 112 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 700 }[/math] [math]\displaystyle{ 116 }[/math] [math]\displaystyle{ 2688 }[/math] [math]\displaystyle{ \frac{14368}{3} }[/math] [math]\displaystyle{ \frac{2560}{3} }[/math] [math]\displaystyle{ 688 }[/math] [math]\displaystyle{ 2304 }[/math] [math]\displaystyle{ 6272 }[/math] [math]\displaystyle{ 16800 }[/math] [math]\displaystyle{ 2784 }[/math] [math]\displaystyle{ \frac{166191}{5} }[/math] [math]\displaystyle{ \frac{1876}{5} }[/math] [math]\displaystyle{ \frac{207244}{15} }[/math] [math]\displaystyle{ \frac{689}{3} }[/math] [math]\displaystyle{ \frac{9391}{5} }[/math]

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]4 is the signature of 9 49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 01234567χ
19       2-2
17      1 1
15     32 -1
13    21  1
11   23   1
9  22    0
7  2     2
512      -1
31       1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math] 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=
Crossings[Knot[9, 49]]
Out[2]=  
9
In[3]:=
PD[Knot[9, 49]]
Out[3]=  
PD[X[6, 2, 7, 1], X[12, 8, 13, 7], X[5, 15, 6, 14], X[3, 11, 4, 10], 

 X[11, 3, 12, 2], X[15, 5, 16, 4], X[17, 9, 18, 8], X[9, 17, 10, 16], 



  X[18, 14, 1, 13]]
In[4]:=
GaussCode[Knot[9, 49]]
Out[4]=  
GaussCode[1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9]
In[5]:=
BR[Knot[9, 49]]
Out[5]=  
BR[4, {1, 1, 2, 1, 1, -3, 2, -1, 2, 3, 3}]
In[6]:=
alex = Alexander[Knot[9, 49]][t]
Out[6]=  
    3    6            2

7 + -- - - - 6 t + 3 t



    2   t


    t
In[7]:=
Conway[Knot[9, 49]][z]
Out[7]=  
       2      4
1 + 6 z  + 3 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[9, 49]}
In[9]:=
{KnotDet[Knot[9, 49]], KnotSignature[Knot[9, 49]]}
Out[9]=  
{25, 4}
In[10]:=
J=Jones[Knot[9, 49]][q]
Out[10]=  
 2      3      4      5      6      7      8      9
q  - 2 q  + 4 q  - 4 q  + 5 q  - 4 q  + 3 q  - 2 q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[9, 49]}
In[12]:=
A2Invariant[Knot[9, 49]][q]
Out[12]=  
 6    8    10    14      16    18      20    22    24    26      28
q  - q  + q   + q   + 3 q   + q   + 2 q   - q   - q   - q   - 2 q
In[13]:=
Kauffman[Knot[9, 49]][a, z]
Out[13]=  
                             2        2      2      2      3      3

-3   4    4 z   2 z   2 z   z     10 z    9 z    2 z    3 z    3 z
-- - -- - --- - --- + --- - --- + ----- + ---- - ---- + ---- + ---- - 



8    6    11    9     7     10     8       6      4     11      9



a    a    a     a     a     a      a       a      a     a       a



    3      3      4      4    4    5    5      5    6       6      6
 3 z    3 z    9 z    8 z    z    z    z    2 z    z     4 z    3 z
 ---- - ---- - ---- - ---- + -- - -- + -- + ---- + --- + ---- + ---- + 
   7      5      8      6     4    9    7     5     10     8      6
  a      a      a      a     a    a    a     a     a      a      a

  7    7
 z    z
 -- + --
  9    7


  a    a
In[14]:=
{Vassiliev[2][Knot[9, 49]], Vassiliev[3][Knot[9, 49]]}
Out[14]=  
{0, 14}
In[15]:=
Kh[Knot[9, 49]][q, t]
Out[15]=  
 3    5      5        7  2      9  2      9  3      11  3      11  4

q  + q  + 2 q  t + 2 q  t  + 2 q  t  + 2 q  t  + 2 q   t  + 3 q   t  + 



    13  4    13  5      15  5      15  6    17  6      19  7


  2 q   t  + q   t  + 3 q   t  + 2 q   t  + q   t  + 2 q   t
Retrieved from "https://katlas.org/index.php?title=9_49&oldid=25938"